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vector-toc-level-2"> <a class="vector-toc-link" href="#Millennium_Prize_Problems"> <div class="vector-toc-text"> 1.1 Millennium Prize Problems </div> </a> <ul id="toc-Millennium_Prize_Problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notebooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notebooks"> <div class="vector-toc-text"> 1.2 Notebooks </div> </a> <ul id="toc-Notebooks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unsolved_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Unsolved_problems"> <div class="vector-toc-text"> 2 Unsolved problems </div> </a> <button aria-controls="toc-Unsolved_problems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Unsolved problems subsection </button> <ul id="toc-Unsolved_problems-sublist" class="vector-toc-list"> <li id="toc-Algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebra"> <div class="vector-toc-text"> 2.1 Algebra </div> </a> <ul id="toc-Algebra-sublist" class="vector-toc-list"> <li id="toc-Group_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Group_theory"> <div class="vector-toc-text"> 2.1.1 Group theory </div> </a> <ul id="toc-Group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Representation_theory"> <div class="vector-toc-text"> 2.1.2 Representation theory </div> </a> <ul id="toc-Representation_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analysis"> <div class="vector-toc-text"> 2.2 Analysis </div> </a> <ul id="toc-Analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorics"> <div class="vector-toc-text"> 2.3 Combinatorics </div> </a> <ul id="toc-Combinatorics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamical_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamical_systems"> <div class="vector-toc-text"> 2.4 Dynamical systems </div> </a> <ul id="toc-Dynamical_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Games_and_puzzles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Games_and_puzzles"> <div class="vector-toc-text"> 2.5 Games and puzzles </div> </a> <ul id="toc-Games_and_puzzles-sublist" class="vector-toc-list"> <li id="toc-Combinatorial_games" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Combinatorial_games"> <div class="vector-toc-text"> 2.5.1 Combinatorial games </div> </a> <ul id="toc-Combinatorial_games-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Games_with_imperfect_information" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Games_with_imperfect_information"> <div class="vector-toc-text"> 2.5.2 Games with imperfect information </div> </a> <ul id="toc-Games_with_imperfect_information-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> 2.6 Geometry </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> <li id="toc-Algebraic_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Algebraic_geometry"> <div class="vector-toc-text"> 2.6.1 Algebraic geometry </div> </a> <ul id="toc-Algebraic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covering_and_packing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Covering_and_packing"> <div class="vector-toc-text"> 2.6.2 Covering and packing </div> </a> <ul id="toc-Covering_and_packing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Differential_geometry"> <div class="vector-toc-text"> 2.6.3 Differential geometry </div> </a> <ul id="toc-Differential_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discrete_geometry"> <div class="vector-toc-text"> 2.6.4 Discrete geometry </div> </a> <ul id="toc-Discrete_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_geometry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Euclidean_geometry"> <div class="vector-toc-text"> 2.6.5 Euclidean geometry </div> </a> <ul id="toc-Euclidean_geometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graph_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graph_theory"> <div class="vector-toc-text"> 2.7 Graph theory </div> </a> <ul id="toc-Graph_theory-sublist" class="vector-toc-list"> <li id="toc-Algebraic_graph_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Algebraic_graph_theory"> <div class="vector-toc-text"> 2.7.1 Algebraic graph theory </div> </a> <ul id="toc-Algebraic_graph_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Games_on_graphs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Games_on_graphs"> <div class="vector-toc-text"> 2.7.2 Games on graphs </div> </a> <ul id="toc-Games_on_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graph_coloring_and_labeling" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Graph_coloring_and_labeling"> <div class="vector-toc-text"> 2.7.3 Graph coloring and labeling </div> </a> <ul id="toc-Graph_coloring_and_labeling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graph_drawing_and_embedding" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Graph_drawing_and_embedding"> <div class="vector-toc-text"> 2.7.4 Graph drawing and embedding </div> </a> <ul id="toc-Graph_drawing_and_embedding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Restriction_of_graph_parameters" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Restriction_of_graph_parameters"> <div class="vector-toc-text"> 2.7.5 Restriction of graph parameters </div> </a> <ul id="toc-Restriction_of_graph_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgraphs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Subgraphs"> <div class="vector-toc-text"> 2.7.6 Subgraphs </div> </a> <ul id="toc-Subgraphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Word-representation_of_graphs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Word-representation_of_graphs"> <div class="vector-toc-text"> 2.7.7 Word-representation of graphs </div> </a> <ul id="toc-Word-representation_of_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous_graph_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Miscellaneous_graph_theory"> <div class="vector-toc-text"> 2.7.8 Miscellaneous graph theory </div> </a> <ul id="toc-Miscellaneous_graph_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Model_theory_and_formal_languages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Model_theory_and_formal_languages"> <div class="vector-toc-text"> 2.8 Model theory and formal languages </div> </a> <ul id="toc-Model_theory_and_formal_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_theory"> <div class="vector-toc-text"> 2.9 Probability theory </div> </a> <ul id="toc-Probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> 2.10 Number theory </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> <li id="toc-General" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> 2.10.1 General </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additive_number_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Additive_number_theory"> <div class="vector-toc-text"> 2.10.2 Additive number theory </div> </a> <ul id="toc-Additive_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_number_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Algebraic_number_theory"> <div class="vector-toc-text"> 2.10.3 Algebraic number theory </div> </a> <ul id="toc-Algebraic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_number_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Computational_number_theory"> <div class="vector-toc-text"> 2.10.4 Computational number theory </div> </a> <ul id="toc-Computational_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diophantine_approximation_and_transcendental_number_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Diophantine_approximation_and_transcendental_number_theory"> <div class="vector-toc-text"> 2.10.5 Diophantine approximation and transcendental number theory </div> </a> <ul id="toc-Diophantine_approximation_and_transcendental_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diophantine_equations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Diophantine_equations"> <div class="vector-toc-text"> 2.10.6 Diophantine equations </div> </a> <ul id="toc-Diophantine_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Prime_numbers"> <div class="vector-toc-text"> 2.10.7 Prime numbers </div> </a> <ul id="toc-Prime_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Set_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set_theory"> <div class="vector-toc-text"> 2.11 Set theory </div> </a> <ul id="toc-Set_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> 2.12 Topology </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Problems_solved_since_1995" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Problems_solved_since_1995"> <div class="vector-toc-text"> 3 Problems solved since 1995 </div> </a> <button aria-controls="toc-Problems_solved_since_1995-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Problems solved since 1995 subsection </button> <ul id="toc-Problems_solved_since_1995-sublist" class="vector-toc-list"> <li id="toc-Algebra_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebra_2"> <div class="vector-toc-text"> 3.1 Algebra </div> </a> <ul id="toc-Algebra_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analysis_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analysis_2"> <div class="vector-toc-text"> 3.2 Analysis </div> </a> <ul id="toc-Analysis_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorics_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorics_2"> <div class="vector-toc-text"> 3.3 Combinatorics </div> </a> <ul id="toc-Combinatorics_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamical_systems_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamical_systems_2"> <div class="vector-toc-text"> 3.4 Dynamical systems </div> </a> <ul id="toc-Dynamical_systems_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Game_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Game_theory"> <div class="vector-toc-text"> 3.5 Game theory </div> </a> <ul id="toc-Game_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometry_2"> <div class="vector-toc-text"> 3.6 Geometry </div> </a> <ul id="toc-Geometry_2-sublist" class="vector-toc-list"> <li id="toc-21st_century" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#21st_century"> <div class="vector-toc-text"> 3.6.1 21st century </div> </a> <ul id="toc-21st_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-20th_century" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#20th_century"> <div class="vector-toc-text"> 3.6.2 20th century </div> </a> <ul id="toc-20th_century-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graph_theory_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graph_theory_2"> <div class="vector-toc-text"> 3.7 Graph theory </div> </a> <ul id="toc-Graph_theory_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_theory_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_theory_2"> <div class="vector-toc-text"> 3.8 Group theory </div> </a> <ul id="toc-Group_theory_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory_2"> <div class="vector-toc-text"> 3.9 Number theory </div> </a> <ul id="toc-Number_theory_2-sublist" class="vector-toc-list"> <li id="toc-21st_century_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#21st_century_2"> <div class="vector-toc-text"> 3.9.1 21st century </div> </a> <ul id="toc-21st_century_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-20th_century_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#20th_century_2"> <div class="vector-toc-text"> 3.9.2 20th century </div> </a> <ul id="toc-20th_century_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ramsey_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ramsey_theory"> <div class="vector-toc-text"> 3.10 Ramsey theory </div> </a> <ul id="toc-Ramsey_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theoretical_computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theoretical_computer_science"> <div class="vector-toc-text"> 3.11 Theoretical computer science </div> </a> <ul id="toc-Theoretical_computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology_2"> <div class="vector-toc-text"> 3.12 Topology </div> </a> <ul id="toc-Topology_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncategorised" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uncategorised"> <div class="vector-toc-text"> 3.13 Uncategorised </div> </a> <ul id="toc-Uncategorised-sublist" class="vector-toc-list"> <li id="toc-2010s" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#2010s"> <div class="vector-toc-text"> 3.13.1 2010s </div> </a> <ul id="toc-2010s-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2000s" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#2000s"> <div class="vector-toc-text"> 3.13.2 2000s </div> </a> <ul id="toc-2000s-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 4 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 5 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 6 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 7 Further reading </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further reading subsection </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Books_discussing_problems_solved_since_1995" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books_discussing_problems_solved_since_1995"> <div class="vector-toc-text"> 7.1 Books discussing problems solved since 1995 </div> </a> <ul id="toc-Books_discussing_problems_solved_since_1995-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books_discussing_unsolved_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books_discussing_unsolved_problems"> <div class="vector-toc-text"> 7.2 Books discussing unsolved problems </div> </a> <ul id="toc-Books_discussing_unsolved_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 8 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">List of unsolved problems in mathematics</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 28 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-28" aria-hidden="true" > 28 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%A6%D9%85%D8%A9_%D8%A7%D9%84%D9%85%D8%B3%D8%A7%D8%A6%D9%84_%D8%BA%D9%8A%D8%B1_%D8%A7%D9%84%D9%85%D8%AD%D9%84%D9%88%D9%84%D8%A9_%D9%81%D9%8A_%D8%A7%D9%84%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="قائمة المسائل غير المحلولة في الرياضيات – Arabic" lang="ar" hreflang="ar" data-title="قائمة المسائل غير المحلولة في الرياضيات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li 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You can help by <a href="/wiki/Special:EditPage/List_of_unsolved_problems_in_mathematics" title="Special:EditPage/List of unsolved problems in mathematics">adding missing items</a> with <a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">reliable sources</a>.</div> Many <a href="/wiki/Mathematical_problems" class="mw-redirect" title="Mathematical problems">mathematical problems</a> have been stated but not yet solved. These problems come from many <a href="/wiki/Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">areas of mathematics</a>, such as <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a>, <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic</a>, <a href="/wiki/Differential_geometry" title="Differential geometry">differential</a>, <a href="/wiki/Discrete_geometry" title="Discrete geometry">discrete</a> and <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometries</a>, <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <a href="/wiki/Model_theory" title="Model theory">model theory</a>, <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Set_theory" title="Set theory">set theory</a>, <a href="/wiki/Ramsey_theory" title="Ramsey theory">Ramsey theory</a>, <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>, and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Lists_of_unsolved_problems_in_mathematics">Lists of unsolved problems in mathematics</h2></div> Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions. <table class="wikitable sortable"> <tbody><tr> <th>List</th> <th>Number of problems</th> <th>Number unsolved or incompletely solved</th> <th>Proposed by</th> <th>Proposed in </th></tr> <tr> <td><a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">Hilbert's problems</a><a href="#cite_note-1">[1]</a></td> <td>23</td> <td>15</td> <td><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a></td> <td>1900 </td></tr> <tr> <td><a href="/wiki/Landau%27s_problems" title="Landau's problems">Landau's problems</a><a href="#cite_note-2">[2]</a></td> <td>4</td> <td>4</td> <td><a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a></td> <td>1912 </td></tr> <tr> <td><a href="/w/index.php?title=Taniyama%27s_problems&action=edit&redlink=1" class="new" title="Taniyama's problems (page does not exist)">Taniyama's problems</a><a href="#cite_note-3">[3]</a></td> <td>36</td> <td>-</td> <td><a href="/wiki/Yutaka_Taniyama" title="Yutaka Taniyama">Yutaka Taniyama</a></td> <td>1955 </td></tr> <tr> <td><a href="/w/index.php?title=Thurston%27s_24_questions&action=edit&redlink=1" class="new" title="Thurston's 24 questions (page does not exist)">Thurston's 24 questions</a><a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a></td> <td>24</td> <td>-</td> <td><a href="/wiki/William_Thurston" title="William Thurston">William Thurston</a></td> <td>1982 </td></tr> <tr> <td><a href="/wiki/Smale%27s_problems" title="Smale's problems">Smale's problems</a></td> <td>18</td> <td>14</td> <td><a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a></td> <td>1998 </td></tr> <tr> <td><a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a></td> <td>7</td> <td>6<a href="#cite_note-auto1-6">[6]</a></td> <td><a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a></td> <td>2000 </td></tr> <tr> <td><a href="/wiki/Simon_problems" title="Simon problems">Simon problems</a></td> <td>15</td> <td><12<a href="#cite_note-7">[7]</a><a href="#cite_note-guardian-8">[8]</a></td> <td><a href="/wiki/Barry_Simon" title="Barry Simon">Barry Simon</a></td> <td>2000 </td></tr> <tr> <td><a href="/w/index.php?title=Unsolved_Problems_on_Mathematics_for_the_21st_Century&action=edit&redlink=1" class="new" title="Unsolved Problems on Mathematics for the 21st Century (page does not exist)">Unsolved Problems on Mathematics for the 21st Century</a><a href="#cite_note-9">[9]</a></td> <td>22</td> <td>-</td> <td>Jair Minoro Abe, Shotaro Tanaka</td> <td>2001 </td></tr> <tr> <td><a href="/wiki/DARPA" title="DARPA">DARPA</a>'s math challenges<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a></td> <td>23</td> <td>-</td> <td><a href="/wiki/DARPA" title="DARPA">DARPA</a></td> <td>2007 </td></tr> <tr> <td><a href="/w/index.php?title=Erd%C5%91s%27s_problems&action=edit&redlink=1" class="new" title="Erdős's problems (page does not exist)">Erdős's problems</a><a href="#cite_note-12">[12]</a></td> <td>>860</td> <td>580</td> <td><a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a></td> <td>Over six decades of Erdős' career, from the 1930s to 1990s </td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Riemann-Zeta-Func.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Riemann-Zeta-Func.png/250px-Riemann-Zeta-Func.png" decoding="async" width="250" height="332" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Riemann-Zeta-Func.png/375px-Riemann-Zeta-Func.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Riemann-Zeta-Func.png/500px-Riemann-Zeta-Func.png 2x" data-file-width="1320" data-file-height="1752" /></a><figcaption>The <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, subject of the celebrated and influential unsolved problem known as the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Millennium_Prize_Problems">Millennium Prize Problems</h3></div> Of the original seven <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a> listed by the <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> in 2000, six remain unsolved to date:<a href="#cite_note-auto1-6">[6]</a> <ul><li><a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></li> <li><a href="/wiki/Hodge_conjecture" title="Hodge conjecture">Hodge conjecture</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_existence_and_smoothness" title="Navier–Stokes existence and smoothness">Navier–Stokes existence and smoothness</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_existence_and_mass_gap" title="Yang–Mills existence and mass gap">Yang–Mills existence and mass gap</a></li></ul> The seventh problem, the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>, was solved by <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a> in 2003.<a href="#cite_note-13">[13]</a> However, a generalization called the <a href="/wiki/Generalized_Poincar%C3%A9_conjecture" title="Generalized Poincaré conjecture">smooth four-dimensional Poincaré conjecture</a>—that is, whether a four-dimensional <a href="/wiki/Topological_sphere" class="mw-redirect" title="Topological sphere">topological sphere</a> can have two or more inequivalent <a href="/wiki/Smooth_structure" title="Smooth structure">smooth structures</a>—is unsolved.<a href="#cite_note-14">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Notebooks">Notebooks</h3></div> <ul><li>The <a href="/wiki/Kourovka,_Sverdlovsk_Oblast" title="Kourovka, Sverdlovsk Oblast">Kourovka</a> Notebook (<a href="/wiki/Russian_language" title="Russian language">Russian</a>: Коуровская тетрадь) is a collection of unsolved problems in <a href="/wiki/Group_theory" title="Group theory">group theory</a>, first published in 1965 and updated many times since.<a href="#cite_note-15">[15]</a></li> <li>The <a href="/wiki/Yekaterinburg" title="Yekaterinburg">Sverdlovsk</a> Notebook (<a href="/wiki/Russian_language" title="Russian language">Russian</a>: Свердловская тетрадь) is a collection of unsolved problems in <a href="/wiki/Semigroup_theory" class="mw-redirect" title="Semigroup theory">semigroup theory</a>, first published in 1965 and updated every 2 to 4 years since.<a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a></li> <li>The <a href="/wiki/Dniester" title="Dniester">Dniester</a> Notebook (<a href="/wiki/Russian_language" title="Russian language">Russian</a>: Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a> and <a href="/wiki/Modulus_(algebraic_number_theory)" title="Modulus (algebraic number theory)">modulus theory</a>.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a></li> <li>The <a href="/w/index.php?title=Erlagol&action=edit&redlink=1" class="new" title="Erlagol (page does not exist)">Erlagol</a> Notebook (<a href="/wiki/Russian_language" title="Russian language">Russian</a>: Эрлагольская тетрадь) lists unsolved problems in algebra and <a href="/wiki/Model_theory" title="Model theory">model theory</a>.<a href="#cite_note-21">[21]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Unsolved_problems">Unsolved problems</h2></div> <div class="mw-heading mw-heading3"><h3 id="Algebra">Algebra</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebra" title="Algebra">Algebra</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Regular_tetrahedron_inscribed_in_a_sphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Regular_tetrahedron_inscribed_in_a_sphere.svg/220px-Regular_tetrahedron_inscribed_in_a_sphere.svg.png" decoding="async" width="220" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Regular_tetrahedron_inscribed_in_a_sphere.svg/330px-Regular_tetrahedron_inscribed_in_a_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Regular_tetrahedron_inscribed_in_a_sphere.svg/440px-Regular_tetrahedron_inscribed_in_a_sphere.svg.png 2x" data-file-width="602" data-file-height="532" /></a><figcaption>In the <a href="/wiki/Bloch_sphere" title="Bloch sphere">Bloch sphere</a> representation of a <a href="/wiki/Qubit" title="Qubit">qubit</a>, a <a href="/wiki/SIC-POVM" title="SIC-POVM">SIC-POVM</a> forms a <a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedron</a>. Zauner conjectured that analogous structures exist in complex <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> of all finite dimensions.</figcaption></figure> <ul><li><a href="/wiki/Birch%E2%80%93Tate_conjecture" title="Birch–Tate conjecture">Birch–Tate conjecture</a> on the relation between the order of the <a href="/wiki/Center_(group_theory)" title="Center (group theory)">center</a> of the <a href="/wiki/Steinberg_group_(K-theory)" title="Steinberg group (K-theory)">Steinberg group</a> of the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> of a <a href="/wiki/Algebraic_number_field" title="Algebraic number field">number field</a> to the field's <a href="/wiki/Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta function</a>.</li> <li><a href="/wiki/Bombieri%E2%80%93Lang_conjecture" title="Bombieri–Lang conjecture">Bombieri–Lang conjectures</a> on densities of rational points of <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surfaces</a> and <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> defined on <a href="/wiki/Algebraic_number_field" title="Algebraic number field">number fields</a> and their <a href="/wiki/Field_extension" title="Field extension">field extensions</a>.</li> <li><a href="/wiki/Connes_embedding_problem" title="Connes embedding problem">Connes embedding problem</a> in <a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a> theory</li> <li><a href="/wiki/Crouzeix%27s_conjecture" title="Crouzeix's conjecture">Crouzeix's conjecture</a>: the <a href="/wiki/Matrix_norm" title="Matrix norm">matrix norm</a> of a complex function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> applied to a complex matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is at most twice the <a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">supremum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(z)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(z)|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdbe46515a38cb5b3c75554c463f7ee985853a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.47ex; height:2.843ex;" alt="{\displaystyle |f(z)|}"> over the <a href="/wiki/Numerical_range" title="Numerical range">field of values</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.</li> <li><a href="/wiki/Determinantal_conjecture" title="Determinantal conjecture">Determinantal conjecture</a> on the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the sum of two <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrices</a>.</li> <li><a href="/wiki/Eilenberg%E2%80%93Ganea_conjecture" title="Eilenberg–Ganea conjecture">Eilenberg–Ganea conjecture</a>: a group with <a href="/wiki/Cohomological_dimension" title="Cohomological dimension">cohomological dimension</a> 2 also has a 2-dimensional <a href="/wiki/Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(G,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(G,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ac500e56f9a311b1e02891755822a53a99af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.898ex; height:2.843ex;" alt="{\displaystyle K(G,1)}">.</li> <li><a href="/wiki/Farrell%E2%80%93Jones_conjecture" title="Farrell–Jones conjecture">Farrell–Jones conjecture</a> on whether certain <a href="/wiki/Assembly_map" title="Assembly map">assembly maps</a> are <a href="/wiki/Isomorphisms" class="mw-redirect" title="Isomorphisms">isomorphisms</a>. <ul><li><a href="/wiki/Farrell%E2%80%93Jones_conjecture#Bost_conjecture" title="Farrell–Jones conjecture">Bost conjecture</a>: a specific case of the Farrell–Jones conjecture</li></ul></li> <li><a href="/wiki/Finite_lattice_representation_problem" title="Finite lattice representation problem">Finite lattice representation problem</a>: is every finite <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> isomorphic to the <a href="/wiki/Quotient_(universal_algebra)#Congruence_lattice" title="Quotient (universal algebra)">congruence lattice</a> of some finite <a href="/wiki/Universal_algebra" title="Universal algebra">algebra</a>?<a href="#cite_note-22">[22]</a></li> <li><a href="/wiki/Goncharov_conjecture" title="Goncharov conjecture">Goncharov conjecture</a> on the <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> of certain <a href="/wiki/Motivic_cohomology" title="Motivic cohomology">motivic complexes</a>.</li> <li><a href="/wiki/Green%27s_conjecture" class="mw-redirect" title="Green's conjecture">Green's conjecture</a>: the <a href="/wiki/Clifford%27s_theorem_on_special_divisors" title="Clifford's theorem on special divisors">Clifford index</a> of a non-<a href="/wiki/Hyperelliptic_curve" title="Hyperelliptic curve">hyperelliptic curve</a> is determined by the extent to which it, as a <a href="/wiki/Canonical_bundle" title="Canonical bundle">canonical curve</a>, has <a href="/wiki/Linear_relation" title="Linear relation">linear syzygies</a>.</li> <li><a href="/wiki/Grothendieck%E2%80%93Katz_p-curvature_conjecture" title="Grothendieck–Katz p-curvature conjecture">Grothendieck–Katz p-curvature conjecture</a>: a conjectured <a href="/wiki/Hasse_principle" title="Hasse principle">local–global principle</a> for <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear ordinary differential equations</a>.</li> <li><a href="/wiki/Hadamard_conjecture" class="mw-redirect" title="Hadamard conjecture">Hadamard conjecture</a>: for every positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">, a <a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard matrix</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ea0bbe41a62da2d834f6fcc4298362c11e17ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.374ex; height:2.176ex;" alt="{\displaystyle 4k}"> exists. <ul><li><a href="/wiki/Williamson_conjecture" title="Williamson conjecture">Williamson conjecture</a>: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices.</li></ul></li> <li><a href="/wiki/Hadamard%27s_maximal_determinant_problem" title="Hadamard's maximal determinant problem">Hadamard's maximal determinant problem</a>: what is the largest <a href="/wiki/Determinant" title="Determinant">determinant</a> of a matrix with entries all equal to 1 or –1?</li> <li><a href="/wiki/Hilbert%27s_fifteenth_problem" title="Hilbert's fifteenth problem">Hilbert's fifteenth problem</a>: put <a href="/wiki/Schubert_calculus" title="Schubert calculus">Schubert calculus</a> on a rigorous foundation.</li> <li><a href="/wiki/Hilbert%27s_sixteenth_problem" title="Hilbert's sixteenth problem">Hilbert's sixteenth problem</a>: what are the possible configurations of the <a href="/wiki/Connected_space" title="Connected space">connected components</a> of <a href="/wiki/Harnack%27s_curve_theorem" title="Harnack's curve theorem">M-curves</a>?</li> <li><a href="/wiki/Homological_conjectures_in_commutative_algebra" title="Homological conjectures in commutative algebra">Homological conjectures in commutative algebra</a></li> <li><a href="/wiki/Jacobson%27s_conjecture" title="Jacobson's conjecture">Jacobson's conjecture</a>: the intersection of all powers of the <a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a> of a left-and-right <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian ring</a> is precisely 0.</li> <li><a href="/wiki/Kaplansky%27s_conjectures" title="Kaplansky's conjectures">Kaplansky's conjectures</a></li> <li><a href="/wiki/K%C3%B6the_conjecture" title="Köthe conjecture">Köthe conjecture</a>: if a ring has no <a href="/wiki/Nil_ideal" title="Nil ideal">nil ideal</a> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}">, then it has no nil <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">one-sided ideal</a> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}">.</li> <li><a href="/wiki/Monomial_conjecture" title="Monomial conjecture">Monomial conjecture</a> on <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian</a> <a href="/wiki/Local_ring" title="Local ring">local rings</a></li> <li>Existence of <a href="/wiki/Perfect_cuboid" class="mw-redirect" title="Perfect cuboid">perfect cuboids</a> and associated <a href="/wiki/Cuboid_conjectures" class="mw-redirect" title="Cuboid conjectures">cuboid conjectures</a></li> <li><a href="/wiki/Pierce%E2%80%93Birkhoff_conjecture" title="Pierce–Birkhoff conjecture">Pierce–Birkhoff conjecture</a>: every piecewise-polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab54a3a84df609b7ac88c7f37cc10cbd5e5761fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }"> is the maximum of a finite set of minimums of finite collections of polynomials.</li> <li><a href="/wiki/Rota%27s_basis_conjecture" title="Rota's basis conjecture">Rota's basis conjecture</a>: for matroids of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> disjoint bases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82cda0578ec6b48774c541ecb9bee4a90176e62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle B_{i}}">, it is possible to create an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix whose rows are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82cda0578ec6b48774c541ecb9bee4a90176e62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle B_{i}}"> and whose columns are also bases.</li> <li><a href="/wiki/Serre%27s_conjecture_II_(algebra)" class="mw-redirect" title="Serre's conjecture II (algebra)">Serre's conjecture II</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is a <a href="/wiki/Simply_connected_space" title="Simply connected space">simply connected</a> <a href="/wiki/Semisimple_algebraic_group" class="mw-redirect" title="Semisimple algebraic group">semisimple algebraic group</a> over a perfect <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Cohomological_dimension" title="Cohomological dimension">cohomological dimension</a> at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">, then the <a href="/wiki/Galois_cohomology" title="Galois cohomology">Galois cohomology</a> set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{1}(F,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{1}(F,G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa9afc3c30f7c4c00e9676d91500924121ef0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.568ex; height:3.176ex;" alt="{\displaystyle H^{1}(F,G)}"> is zero.</li> <li><a href="/wiki/Serre%27s_multiplicity_conjectures" title="Serre's multiplicity conjectures">Serre's positivity conjecture</a> that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is a commutative <a href="/wiki/Regular_local_ring" title="Regular local ring">regular local ring</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P,Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>,</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P,Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b3c5b21ee2d8a12253a952e4c4b1733ceb0735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.618ex; height:2.509ex;" alt="{\displaystyle P,Q}"> are <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/474abd9433fba154199dffc7a82f5d56643470a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.193ex; height:2.843ex;" alt="{\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)}"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (R/P,R/Q)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mo>,</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (R/P,R/Q)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d319b0b57fc8e393df931c5910d7c0b6ef1f2523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.996ex; height:2.843ex;" alt="{\displaystyle \chi (R/P,R/Q)>0}">.</li> <li><a href="/wiki/Uniform_boundedness_conjecture_for_rational_points" title="Uniform boundedness conjecture for rational points">Uniform boundedness conjecture for rational points</a>: do <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a> of <a href="/wiki/Geometric_genus" title="Geometric genus">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84b2394f64ae9927e09ee51b848be213edecafe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}"> over <a href="/wiki/Algebraic_number_field" title="Algebraic number field">number fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> have at most some bounded number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(K,g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(K,g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc13b3e0400ce3c69aa0b173b7a5a764d9d25ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.089ex; height:2.843ex;" alt="{\displaystyle N(K,g)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">-<a href="/wiki/Rational_point" title="Rational point">rational points</a>?</li> <li><a href="/wiki/Wild_problem" title="Wild problem">Wild problems</a>: problems involving classification of pairs of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrices under simultaneous conjugation.</li> <li><a href="/wiki/Zariski%E2%80%93Lipman_conjecture" class="mw-redirect" title="Zariski–Lipman conjecture">Zariski–Lipman conjecture</a>: for a <a href="/wiki/Complex_algebraic_variety" title="Complex algebraic variety">complex algebraic variety</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> with <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, if the <a href="/wiki/Derivation_(algebra)" class="mw-redirect" title="Derivation (algebra)">derivations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> are a <a href="/wiki/Free_module" title="Free module">free module</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> is <a href="/wiki/Smooth_algebraic_variety" class="mw-redirect" title="Smooth algebraic variety">smooth</a>.</li> <li>Zauner's conjecture: do <a href="/wiki/SIC-POVM" title="SIC-POVM">SIC-POVMs</a> exist in all dimensions?</li> <li><a href="/wiki/Zilber%E2%80%93Pink_conjecture" title="Zilber–Pink conjecture">Zilber–Pink conjecture</a> that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a mixed <a href="/wiki/Shimura_variety" title="Shimura variety">Shimura variety</a> or <a href="/wiki/Abelian_variety#Semiabelian_variety" title="Abelian variety">semiabelian variety</a> defined over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd13f9c1d91f8036a8c33fb8d6b03a7d08492a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.866ex; height:2.343ex;" alt="{\displaystyle V\subseteq X}"> is a subvariety, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> contains only finitely many atypical subvarieties.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Group_theory">Group theory</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Group_theory" title="Group theory">Group theory</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:FreeBurnsideGroupExp3Gens2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/FreeBurnsideGroupExp3Gens2.png/220px-FreeBurnsideGroupExp3Gens2.png" decoding="async" width="220" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/FreeBurnsideGroupExp3Gens2.png/330px-FreeBurnsideGroupExp3Gens2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/FreeBurnsideGroupExp3Gens2.png/440px-FreeBurnsideGroupExp3Gens2.png 2x" data-file-width="1792" data-file-height="1381" /></a><figcaption>The <a href="/wiki/Free_Burnside_group" class="mw-redirect" title="Free Burnside group">free Burnside group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(2,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(2,3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c9f28d2252e08412db83b4e6432aaa759b2680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.932ex; height:2.843ex;" alt="{\displaystyle B(2,3)}"> is finite; in its <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a>, shown here, each of its 27 elements is represented by a vertex. The question of which other groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(m,n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f1a73c7fa714079c1d7d34ac057cc268407d334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.042ex; height:2.843ex;" alt="{\displaystyle B(m,n)}"> are finite remains open.</figcaption></figure> <ul><li><a href="/wiki/Andrews%E2%80%93Curtis_conjecture" title="Andrews–Curtis conjecture">Andrews–Curtis conjecture</a>: every balanced <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> of the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a> can be transformed into a trivial presentation by a sequence of <a href="/wiki/Nielsen_transformation" title="Nielsen transformation">Nielsen transformations</a> on <a href="/wiki/Presentation_of_a_group#Definition" title="Presentation of a group">relators</a> and conjugations of relators</li> <li><a href="/wiki/Burnside_problem" title="Burnside problem">Burnside problem</a>: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?</li> <li>Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems<a href="#cite_note-23">[23]</a></li> <li><a href="/wiki/Herzog%E2%80%93Sch%C3%B6nheim_conjecture" title="Herzog–Schönheim conjecture">Herzog–Schönheim conjecture</a>: if a finite system of left <a href="/wiki/Coset" title="Coset">cosets</a> of subgroups of a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> form a partition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">, then the finite indices of said subgroups cannot be distinct.</li> <li>The <a href="/wiki/Inverse_Galois_problem" title="Inverse Galois problem">inverse Galois problem</a>: is every finite group the Galois group of a Galois extension of the rationals?</li> <li>Are there an infinite number of <a href="/wiki/Leinster_group" title="Leinster group">Leinster groups</a>?</li> <li>Does <a href="/wiki/Monstrous_moonshine#Generalized_moonshine" title="Monstrous moonshine">generalized moonshine</a> exist?</li> <li>Is every <a href="/wiki/Finitely_presented_group" class="mw-redirect" title="Finitely presented group">finitely presented</a> <a href="/wiki/Periodic_group" class="mw-redirect" title="Periodic group">periodic group</a> finite?</li> <li>Is every group <a href="/wiki/Surjunctive_group" title="Surjunctive group">surjunctive</a>?</li> <li>Is every discrete, countable group <a href="/wiki/Sofic_group" title="Sofic group">sofic</a>?</li> <li><a href="/wiki/Problems_in_loop_theory_and_quasigroup_theory" class="mw-redirect" title="Problems in loop theory and quasigroup theory">Problems in loop theory and quasigroup theory</a> consider generalizations of groups</li></ul> <div class="mw-heading mw-heading4"><h4 id="Representation_theory">Representation theory</h4></div> <ul><li><a href="/wiki/Arthur%27s_conjectures" title="Arthur's conjectures">Arthur's conjectures</a></li> <li><a href="/wiki/Dade%27s_conjecture" title="Dade's conjecture">Dade's conjecture</a> relating the numbers of <a href="/wiki/Character_theory" title="Character theory">characters</a> of <a href="/wiki/Modular_representation_theory#Blocks_and_the_structure_of_the_group_algebra" title="Modular representation theory">blocks</a> of a finite group to the numbers of characters of blocks of local <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>.</li> <li><a href="/wiki/Demazure_conjecture" title="Demazure conjecture">Demazure conjecture</a> on <a href="/wiki/Group_representation" title="Group representation">representations</a> of <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a> over the integers.</li> <li><a href="/wiki/Kazhdan%E2%80%93Lusztig_polynomial#Kazhdan–Lusztig_conjectures" title="Kazhdan–Lusztig polynomial">Kazhdan–Lusztig conjectures</a> relating the values of the <a href="/wiki/Kazhdan%E2%80%93Lusztig_polynomial" title="Kazhdan–Lusztig polynomial">Kazhdan–Lusztig polynomials</a> at 1 with <a href="/wiki/Group_representation" title="Group representation">representations</a> of complex <a href="/wiki/Semisimple_Lie_algebra#Semisimple_and_reductive_groups" title="Semisimple Lie algebra">semisimple Lie groups</a> and <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">Lie algebras</a>.</li> <li><a href="/wiki/McKay_conjecture" title="McKay conjecture">McKay conjecture</a>: in a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">, the number of <a href="/wiki/Character_theory#Definitions" title="Character theory">irreducible complex characters</a> of degree not divisible by a <a href="/wiki/Prime_number" title="Prime number">prime number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is equal to the number of irreducible complex characters of the <a href="/wiki/Centralizer_and_normalizer" title="Centralizer and normalizer">normalizer</a> of any <a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-subgroup</a> within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Analysis">Analysis</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></div> <ul><li>The <a href="/wiki/Brennan_conjecture" title="Brennan conjecture">Brennan conjecture</a>: estimating the integral of powers of the moduli of the derivative of <a href="/wiki/Conformal_map" title="Conformal map">conformal maps</a> into the open unit disk, on certain subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></li> <li><a href="/wiki/Fuglede%27s_conjecture" title="Fuglede's conjecture">Fuglede's conjecture</a> on whether nonconvex sets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> are spectral if and only if they tile by <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a>.</li> <li><a href="/wiki/Goodman%27s_conjecture" title="Goodman's conjecture">Goodman's conjecture</a> on the coefficients of <a href="/wiki/Multivalent_function" class="mw-redirect" title="Multivalent function">multivalent functions</a></li> <li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a> – does every <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operator</a> on a complex <a href="/wiki/Banach_space" title="Banach space">Banach space</a> send some non-trivial <a href="/wiki/Closed_set" title="Closed set">closed</a> subspace to itself?</li> <li>Kung–Traub conjecture on the optimal order of a multipoint iteration without memory<a href="#cite_note-24">[24]</a></li> <li><a href="/wiki/Lehmer%27s_conjecture" title="Lehmer's conjecture">Lehmer's conjecture</a> on the Mahler measure of non-cyclotomic polynomials<a href="#cite_note-25">[25]</a></li> <li>The <a href="/wiki/Mean_value_problem" title="Mean value problem">mean value problem</a>: given a <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f0ac074e1d66eecdf58762164e0afd3d628232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 2}"> and a complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, is there a <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367a795b758b5bcf81ef3920bbc5f8eb26689dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.929ex; height:3.009ex;" alt="{\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|}">?</li> <li>The <a href="/wiki/Pompeiu_problem" title="Pompeiu problem">Pompeiu problem</a> on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy<a href="#cite_note-26">[26]</a></li> <li><a href="/wiki/Sendov%27s_conjecture" title="Sendov's conjecture">Sendov's conjecture</a>: if a complex polynomial with degree at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> has all roots in the closed <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a>, then each root is within distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> from some <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a>.</li> <li><a href="/wiki/Analytic_capacity#Vitushkin's_conjecture" title="Analytic capacity">Vitushkin's conjecture</a> on compact subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> with <a href="/wiki/Analytic_capacity" title="Analytic capacity">analytic capacity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></li> <li>What is the exact value of <a href="/wiki/Landau%27s_constants" class="mw-redirect" title="Landau's constants">Landau's constants</a>, including <a href="/wiki/Bloch%27s_theorem_(complex_variables)#Bloch's_and_Landau's_constants" class="mw-redirect" title="Bloch's theorem (complex variables)">Bloch's constant</a>?</li></ul> <ul><li>Regularity of solutions of <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equations</a></li> <li>Convergence of <a href="/wiki/Flint_Hills_series" class="mw-redirect" title="Flint Hills series">Flint Hills series</a></li> <li>Regularity of solutions of <a href="/wiki/Vlasov%E2%80%93Maxwell_equations" class="mw-redirect" title="Vlasov–Maxwell equations">Vlasov–Maxwell equations</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Combinatorics">Combinatorics</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></div> <ul><li>The <a href="/wiki/1/3%E2%80%932/3_conjecture" title="1/3–2/3 conjecture">1/3–2/3 conjecture</a> – does every finite <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> that is not <a href="/wiki/Totally_ordered" class="mw-redirect" title="Totally ordered">totally ordered</a> contain two elements x and y such that the probability that x appears before y in a random <a href="/wiki/Linear_extension" title="Linear extension">linear extension</a> is between 1/3 and 2/3?<a href="#cite_note-27">[27]</a></li> <li>The <a href="/wiki/Dittert_conjecture" title="Dittert conjecture">Dittert conjecture</a> concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition</li> <li><a href="/wiki/Problems_in_Latin_squares" title="Problems in Latin squares">Problems in Latin squares</a> – open questions concerning <a href="/wiki/Latin_squares" class="mw-redirect" title="Latin squares">Latin squares</a></li> <li>The <a href="/wiki/Lonely_runner_conjecture" title="Lonely runner conjecture">lonely runner conjecture</a> – if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e9fedad8c70c6331b2640b56c23cef8c884e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.536ex; height:2.843ex;" alt="{\displaystyle 1/k}"> from each other runner) at some time?<a href="#cite_note-28">[28]</a></li> <li><a href="/wiki/Map_folding" title="Map folding">Map folding</a> – various problems in map folding and stamp folding.</li> <li><a href="/wiki/No-three-in-line_problem" title="No-three-in-line problem">No-three-in-line problem</a> – how many points can be placed in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> grid so that no three of them lie on a line?</li> <li><a href="/wiki/Rudin%27s_conjecture" title="Rudin's conjecture">Rudin's conjecture</a> on the number of squares in finite <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progressions</a><a href="#cite_note-29">[29]</a></li> <li>The <a href="/wiki/Sunflower_(mathematics)" title="Sunflower (mathematics)">sunflower conjecture</a> – can the number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> size sets required for the existence of a sunflower of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> sets be bounded by an exponential function in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> for every fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b91ae29874a80d658689e051217e63068e49708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>2}">?</li> <li>Frankl's <a href="/wiki/Union-closed_sets_conjecture" title="Union-closed sets conjecture">union-closed sets conjecture</a> – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets<a href="#cite_note-30">[30]</a></li></ul> <ul><li>Give a combinatorial interpretation of the <a href="/wiki/Kronecker_coefficient" title="Kronecker coefficient">Kronecker coefficients</a><a href="#cite_note-31">[31]</a></li> <li>The values of the <a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c856803491e153263b91aacc660e26ef129d467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle M(n)}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20b8a0ac431041fa3ed940ebccbb54e9a8b332f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.818ex; height:2.343ex;" alt="{\displaystyle n\geq 10}"><a href="#cite_note-32">[32]</a></li> <li>The values of the <a href="/wiki/Ramsey_numbers" class="mw-redirect" title="Ramsey numbers">Ramsey numbers</a>, particularly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(5,5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(5,5)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce54825efec1236758d3393b148f0e1d92788b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.932ex; height:2.843ex;" alt="{\displaystyle R(5,5)}"></li> <li>The values of the <a href="/wiki/Van_der_Waerden_number" title="Van der Waerden number">Van der Waerden numbers</a></li> <li>Finding a function to model n-step <a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">self-avoiding walks</a><a href="#cite_note-33">[33]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Dynamical_systems">Dynamical systems</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical system</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mandel_zoom_07_satellite.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Mandel_zoom_07_satellite.jpg/220px-Mandel_zoom_07_satellite.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Mandel_zoom_07_satellite.jpg/330px-Mandel_zoom_07_satellite.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Mandel_zoom_07_satellite.jpg/440px-Mandel_zoom_07_satellite.jpg 2x" data-file-width="2560" data-file-height="1920" /></a><figcaption>A detail of the <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a>. It is not known whether the Mandelbrot set is <a href="/wiki/Locally_connected_space" title="Locally connected space">locally connected</a> or not.</figcaption></figure> <ul><li><a href="/wiki/Arnold%E2%80%93Givental_conjecture" class="mw-redirect" title="Arnold–Givental conjecture">Arnold–Givental conjecture</a> and <a href="/wiki/Symplectomorphism#Arnold_conjecture" title="Symplectomorphism">Arnold conjecture</a> – relating symplectic geometry to Morse theory.</li> <li><a href="/wiki/Quantum_chaos#Berry–Tabor_conjecture" title="Quantum chaos">Berry–Tabor conjecture</a> in <a href="/wiki/Quantum_chaos" title="Quantum chaos">quantum chaos</a></li> <li><a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach's</a> problem – is there an <a href="/wiki/Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">ergodic system</a> with simple Lebesgue spectrum?<a href="#cite_note-34">[34]</a></li> <li><a href="/wiki/George_David_Birkhoff" title="George David Birkhoff">Birkhoff</a> conjecture – if a <a href="/wiki/Dynamical_billiards" title="Dynamical billiards">billiard table</a> is strictly convex and integrable, is its boundary necessarily an ellipse?<a href="#cite_note-35">[35]</a></li> <li><a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a> (aka the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1623e0fa64e26934ce78d9f779af38a90de4d157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 3n+1}"> conjecture)</li> <li><a href="/wiki/Eden%27s_conjecture" title="Eden's conjecture">Eden's conjecture</a> that the <a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">supremum</a> of the local <a href="/wiki/Lyapunov_dimension" title="Lyapunov dimension">Lyapunov dimensions</a> on the global <a href="/wiki/Attractor" title="Attractor">attractor</a> is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.</li> <li><a href="/wiki/Alexandre_Eremenko" title="Alexandre Eremenko">Eremenko's</a> conjecture: every component of the <a href="/wiki/Escaping_set" title="Escaping set">escaping set</a> of an <a href="/wiki/Entire_function" title="Entire function">entire</a> <a href="/wiki/Transcendental_function" title="Transcendental function">transcendental</a> function is unbounded.</li> <li><a href="/wiki/Fatou_conjecture" title="Fatou conjecture">Fatou conjecture</a> that a quadratic family of maps from the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> to itself is hyperbolic for an open dense set of parameters.</li> <li><a href="/wiki/Hillel_Furstenberg" title="Hillel Furstenberg">Furstenberg</a> conjecture – is every invariant and <a href="/wiki/Ergodicity" title="Ergodicity">ergodic</a> measure for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times 2,\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> <mn>2</mn> <mo>,</mo> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times 2,\times 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ff8bc142eaf3ce64a03a5c1b79c9f4dc3bd2d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.975ex; height:2.509ex;" alt="{\displaystyle \times 2,\times 3}"> action on the circle either Lebesgue or atomic?</li> <li><a href="/wiki/Kaplan%E2%80%93Yorke_conjecture" title="Kaplan–Yorke conjecture">Kaplan–Yorke conjecture</a> on the dimension of an <a href="/wiki/Attractor" title="Attractor">attractor</a> in terms of its <a href="/wiki/Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponents</a></li> <li><a href="/wiki/Grigory_Margulis" title="Grigory Margulis">Margulis</a> conjecture – measure classification for diagonalizable actions in higher-rank groups.</li> <li><a href="/wiki/MLC_conjecture" class="mw-redirect" title="MLC conjecture">MLC conjecture</a> – is the Mandelbrot set locally connected?</li> <li>Many problems concerning an <a href="/wiki/Outer_billiard#open_problems" class="mw-redirect" title="Outer billiard">outer billiard</a>, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.</li> <li>Quantum unique ergodicity conjecture on the distribution of large-frequency <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a> on a <a href="/wiki/Curvature" title="Curvature">negatively-curved</a> <a href="/wiki/Manifold" title="Manifold">manifold</a><a href="#cite_note-36">[36]</a></li> <li><a href="/wiki/Vladimir_Abramovich_Rokhlin" title="Vladimir Abramovich Rokhlin">Rokhlin's</a> multiple mixing problem – are all <a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">strongly mixing</a> systems also strongly 3-mixing?<a href="#cite_note-37">[37]</a></li> <li><a href="/wiki/Weinstein_conjecture" title="Weinstein conjecture">Weinstein conjecture</a> – does a regular compact <a href="/wiki/Contact_type" title="Contact type">contact type</a> <a href="/wiki/Level_set" title="Level set">level set</a> of a <a href="/wiki/Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian</a> on a <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> carry at least one periodic orbit of the Hamiltonian flow?</li></ul> <ul><li>Does every positive integer generate a <a href="/wiki/Juggler_sequence" title="Juggler sequence">juggler sequence</a> terminating at 1?</li> <li><a href="/wiki/Lyapunov_function" title="Lyapunov function">Lyapunov function: Lyapunov's second method for stability</a> – For what classes of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ODEs</a>, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?</li> <li>Is every <a href="/wiki/Reversible_cellular_automaton" title="Reversible cellular automaton">reversible cellular automaton</a> in three or more dimensions locally reversible?<a href="#cite_note-38">[38]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Games_and_puzzles">Games and puzzles</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Game_theory" title="Game theory">Game theory</a></div> <div class="mw-heading mw-heading4"><h4 id="Combinatorial_games">Combinatorial games</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a></div> <ul><li><a href="/wiki/Sudoku" title="Sudoku">Sudoku</a>: <ul><li>How many puzzles have exactly one solution?<a href="#cite_note-openq-39">[39]</a></li> <li>How many puzzles with exactly one solution are <a href="/wiki/Glossary_of_Sudoku#Other_terminology" title="Glossary of Sudoku">minimal</a>?<a href="#cite_note-openq-39">[39]</a></li> <li>What is the <a href="/wiki/Mathematics_of_Sudoku#Maximum_number_of_givens" title="Mathematics of Sudoku">maximum number of givens</a> for a <a href="/wiki/Glossary_of_Sudoku#Other_terminology" title="Glossary of Sudoku">minimal</a> puzzle?<a href="#cite_note-openq-39">[39]</a></li></ul></li> <li><a href="/wiki/Tic-tac-toe_variants" title="Tic-tac-toe variants">Tic-tac-toe variants</a>: <ul><li>Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also <a href="/wiki/Hales%E2%80%93Jewett_theorem" title="Hales–Jewett theorem">Hales–Jewett theorem</a> and <a href="/wiki/Nd_game" title="Nd game">nd game</a>)<a href="#cite_note-40">[40]</a></li></ul></li> <li><a href="/wiki/Chess" title="Chess">Chess</a>: <ul><li>What is the outcome of a perfectly played game of chess? (See also <a href="/wiki/First-move_advantage_in_chess" title="First-move advantage in chess">first-move advantage in chess</a>)</li></ul></li> <li><a href="/wiki/Go_(game)" title="Go (game)">Go</a>: <ul><li>What is the perfect value of <a href="/wiki/Komi_(Go)" title="Komi (Go)">Komi</a>?</li></ul></li> <li>Are the nim-sequences of all finite <a href="/wiki/Octal_game" title="Octal game">octal games</a> eventually periodic?</li> <li>Is the nim-sequence of <a href="/wiki/Grundy%27s_game" title="Grundy's game">Grundy's game</a> eventually periodic?</li></ul> <div class="mw-heading mw-heading4"><h4 id="Games_with_imperfect_information">Games with imperfect information</h4></div> <ul><li><a href="/wiki/Rendezvous_problem" title="Rendezvous problem">Rendezvous problem</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Geometry">Geometry</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Geometry" title="Geometry">Geometry</a></div> <div class="mw-heading mw-heading4"><h4 id="Algebraic_geometry">Algebraic geometry</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a></div> <ul><li><a href="/wiki/Abundance_conjecture" title="Abundance conjecture">Abundance conjecture</a>: if the <a href="/wiki/Canonical_bundle" title="Canonical bundle">canonical bundle</a> of a <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> with <a href="/wiki/Canonical_singularity#Pairs" title="Canonical singularity">Kawamata log terminal singularities</a> is <a href="/wiki/Nef_line_bundle" title="Nef line bundle">nef</a>, then it is semiample.</li> <li><a href="/wiki/Bass_conjecture" title="Bass conjecture">Bass conjecture</a> on the <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finite generation</a> of certain <a href="/wiki/Algebraic_K-theory" title="Algebraic K-theory">algebraic K-groups</a>.</li> <li><a href="/wiki/Bass%E2%80%93Quillen_conjecture" title="Bass–Quillen conjecture">Bass–Quillen conjecture</a> relating <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> over a <a href="/wiki/Regular_local_ring#Regular_ring" title="Regular local ring">regular</a> <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian ring</a> and over the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A[t_{1},\ldots ,t_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[t_{1},\ldots ,t_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db7c2e068068f81d1be4e7ed8f140f0f69fea50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.167ex; height:2.843ex;" alt="{\displaystyle A[t_{1},\ldots ,t_{n}]}">.</li> <li><a href="/wiki/Deligne_conjecture" class="mw-redirect" title="Deligne conjecture">Deligne conjecture</a>: any one of numerous named for <a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Pierre Deligne</a>. <ul><li><a href="/wiki/Deligne%27s_conjecture_on_Hochschild_cohomology" title="Deligne's conjecture on Hochschild cohomology">Deligne's conjecture on Hochschild cohomology</a> about the <a href="/wiki/Operad" title="Operad">operadic</a> structure on <a href="/wiki/Hochschild_homology" title="Hochschild homology">Hochschild cochain complex</a>.</li></ul></li> <li><a href="/wiki/Dixmier_conjecture" title="Dixmier conjecture">Dixmier conjecture</a>: any <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> of a <a href="/wiki/Weyl_algebra" title="Weyl algebra">Weyl algebra</a> is an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a>.</li> <li><a href="/wiki/Fr%C3%B6berg_conjecture" title="Fröberg conjecture">Fröberg conjecture</a> on the <a href="/wiki/Hilbert_series_and_Hilbert_polynomial" title="Hilbert series and Hilbert polynomial">Hilbert functions</a> of a set of forms.</li> <li><a href="/wiki/Fujita_conjecture" title="Fujita conjecture">Fujita conjecture</a> regarding the line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{M}\otimes L^{\otimes m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>⊗</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{M}\otimes L^{\otimes m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ac21e9e6433e387fdf9efc5a112c2a9fdbe98d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.309ex; height:2.843ex;" alt="{\displaystyle K_{M}\otimes L^{\otimes m}}"> constructed from a <a href="/wiki/Positive_form#Positive_line_bundles" title="Positive form">positive</a> <a href="/wiki/Holomorphic_vector_bundle" title="Holomorphic vector bundle">holomorphic line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> on a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> and the <a href="/wiki/Canonical_bundle" title="Canonical bundle">canonical line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14296e7a6cf6d829e6dd8bbb40421855edc4de67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.932ex; height:2.509ex;" alt="{\displaystyle K_{M}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></li> <li><a href="/wiki/General_elephant" title="General elephant">General elephant problem</a>: do <a href="/wiki/General_elephant" title="General elephant">general elephants</a> have at most <a href="/wiki/Du_Val_singularity" title="Du Val singularity">Du Val singularities</a>?</li> <li>Hartshorne's conjectures<a href="#cite_note-41">[41]</a></li> <li><a href="/wiki/Jacobian_conjecture" title="Jacobian conjecture">Jacobian conjecture</a>: if a <a href="/wiki/Polynomial_mapping" title="Polynomial mapping">polynomial mapping</a> over a <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a>-0 field has a constant nonzero <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian determinant</a>, then it has a <a href="/wiki/Morphism_of_algebraic_varieties" title="Morphism of algebraic varieties">regular</a> (i.e. with polynomial components) inverse function.</li> <li><a href="/wiki/Manin_conjecture" title="Manin conjecture">Manin conjecture</a> on the distribution of <a href="/wiki/Rational_point" title="Rational point">rational points</a> of bounded <a href="/wiki/Height_function" title="Height function">height</a> in certain subsets of <a href="/wiki/Fano_variety" title="Fano variety">Fano varieties</a></li> <li><a href="/wiki/Maulik%E2%80%93Nekrasov%E2%80%93Okounkov%E2%80%93Pandharipande_conjecture" class="mw-redirect" title="Maulik–Nekrasov–Okounkov–Pandharipande conjecture">Maulik–Nekrasov–Okounkov–Pandharipande conjecture</a> on an equivalence between <a href="/wiki/Gromov%E2%80%93Witten_invariant" title="Gromov–Witten invariant">Gromov–Witten theory</a> and <a href="/wiki/Donaldson%E2%80%93Thomas_theory" title="Donaldson–Thomas theory">Donaldson–Thomas theory</a><a href="#cite_note-42">[42]</a></li> <li><a href="/wiki/Nagata%27s_conjecture_on_curves" title="Nagata's conjecture on curves">Nagata's conjecture on curves</a>, specifically the minimal degree required for a <a href="/wiki/Algebraic_curve" title="Algebraic curve">plane algebraic curve</a> to pass through a collection of very general points with prescribed <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicities</a>.</li> <li><a href="/wiki/Nagata%E2%80%93Biran_conjecture" title="Nagata–Biran conjecture">Nagata–Biran conjecture</a> that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a smooth <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surface</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is an <a href="/wiki/Ample_line_bundle" title="Ample line bundle">ample line bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">, then for sufficiently large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, the <a href="/wiki/Seshadri_constant" title="Seshadri constant">Seshadri constant</a> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>;</mo> <mi>X</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcdcf84dd017bfb73db433eb5420115fb1dbd1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.53ex; height:3.009ex;" alt="{\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}}">.</li> <li><a href="/wiki/Nakai_conjecture" title="Nakai conjecture">Nakai conjecture</a>: if a <a href="/wiki/Complex_algebraic_variety" title="Complex algebraic variety">complex algebraic variety</a> has a ring of <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a> generated by its contained <a href="/wiki/Derivation_(differential_algebra)" title="Derivation (differential algebra)">derivations</a>, then it must be <a href="/wiki/Singular_point_of_an_algebraic_variety" title="Singular point of an algebraic variety">smooth</a>.</li> <li><a href="/wiki/Parshin%27s_conjecture" title="Parshin's conjecture">Parshin's conjecture</a>: the higher <a href="/wiki/Algebraic_K-theory" title="Algebraic K-theory">algebraic K-groups</a> of any <a href="/wiki/Smooth_morphism" title="Smooth morphism">smooth</a> <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> defined over a <a href="/wiki/Finite_field" title="Finite field">finite field</a> must vanish up to torsion.</li> <li><a href="/wiki/Section_conjecture" title="Section conjecture">Section conjecture</a> on splittings of <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphisms</a> from <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental groups</a> of complete <a href="/wiki/Curve#Differential_geometry" title="Curve">smooth curves</a> over finitely-generated <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> to the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">.</li> <li><a href="/wiki/Standard_conjectures" class="mw-redirect" title="Standard conjectures">Standard conjectures</a> on algebraic cycles</li> <li><a href="/wiki/Tate_conjecture" title="Tate conjecture">Tate conjecture</a> on the connection between <a href="/wiki/Algebraic_cycle" title="Algebraic cycle">algebraic cycles</a> on <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> and <a href="/wiki/Galois_module" class="mw-redirect" title="Galois module">Galois representations</a> on <a href="/wiki/%C3%89tale_cohomology" title="Étale cohomology">étale cohomology groups</a>.</li> <li><a href="/wiki/Virasoro_conjecture" title="Virasoro conjecture">Virasoro conjecture</a>: a certain <a href="/wiki/Generating_function" title="Generating function">generating function</a> encoding the <a href="/wiki/Gromov%E2%80%93Witten_invariant" title="Gromov–Witten invariant">Gromov–Witten invariants</a> of a <a href="/wiki/Singular_point_of_an_algebraic_variety" title="Singular point of an algebraic variety">smooth</a> <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> is fixed by an action of half of the <a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a>.</li> <li>Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of <a href="/wiki/Algebraic_variety" title="Algebraic variety">varieties</a> at <a href="/wiki/Singular_point_of_an_algebraic_variety" title="Singular point of an algebraic variety">singular points</a><a href="#cite_note-43">[43]</a></li></ul> <ul><li>Are infinite sequences of <a href="/wiki/Flip_(mathematics)" title="Flip (mathematics)">flips</a> possible in dimensions greater than 3?</li> <li><a href="/wiki/Resolution_of_singularities" title="Resolution of singularities">Resolution of singularities</a> in characteristic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></li></ul> <div class="mw-heading mw-heading4"><h4 id="Covering_and_packing">Covering and packing</h4></div> <ul><li><a href="/wiki/Borsuk%27s_conjecture" title="Borsuk's conjecture">Borsuk's problem</a> on upper and lower bounds for the number of smaller-diameter subsets needed to cover a <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> n-dimensional set.</li> <li>The <a href="/wiki/Covering_problem_of_Rado" title="Covering problem of Rado">covering problem of Rado</a>: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?<a href="#cite_note-44">[44]</a></li> <li>The <a href="/wiki/Circle_packing_in_an_equilateral_triangle" title="Circle packing in an equilateral triangle">Erdős–Oler conjecture</a>: when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is a <a href="/wiki/Triangular_number" title="Triangular number">triangular number</a>, packing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> circles in an equilateral triangle requires a triangle of the same size as packing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> circles<a href="#cite_note-45">[45]</a></li> <li>The <a href="/wiki/Kissing_number_problem" class="mw-redirect" title="Kissing number problem">kissing number problem</a> for dimensions other than 1, 2, 3, 4, 8 and 24<a href="#cite_note-46">[46]</a></li> <li><a href="/wiki/Reinhardt%27s_conjecture" class="mw-redirect" title="Reinhardt's conjecture">Reinhardt's conjecture</a>: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets<a href="#cite_note-47">[47]</a></li> <li><a href="/wiki/Sphere_packing" title="Sphere packing">Sphere packing</a> problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.</li> <li><a href="/wiki/Square_packing_in_a_square" class="mw-redirect" title="Square packing in a square">Square packing in a square</a>: what is the asymptotic growth rate of wasted space?<a href="#cite_note-48">[48]</a></li> <li><a href="/wiki/Ulam%27s_packing_conjecture" title="Ulam's packing conjecture">Ulam's packing conjecture</a> about the identity of the worst-packing convex solid<a href="#cite_note-49">[49]</a></li> <li>The <a href="/wiki/Tammes_problem" title="Tammes problem">Tammes problem</a> for numbers of nodes greater than 14 (except 24).<a href="#cite_note-50">[50]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Differential_geometry">Differential geometry</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></div> <ul><li>The <a href="/wiki/Spherical_Bernstein%27s_problem" title="Spherical Bernstein's problem">spherical Bernstein's problem</a>, a generalization of <a href="/wiki/Bernstein%27s_problem" title="Bernstein's problem">Bernstein's problem</a></li> <li><a href="/wiki/Carath%C3%A9odory_conjecture" title="Carathéodory conjecture">Carathéodory conjecture</a>: any convex, closed, and twice-differentiable surface in three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> admits at least two <a href="/wiki/Umbilical_point" title="Umbilical point">umbilical points</a>.</li> <li><a href="/wiki/Cartan%E2%80%93Hadamard_conjecture" title="Cartan–Hadamard conjecture">Cartan–Hadamard conjecture</a>: can the classical <a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">isoperimetric inequality</a> for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as <a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Cartan–Hadamard manifolds</a>?</li> <li><a href="/wiki/Chern%27s_conjecture_(affine_geometry)" title="Chern's conjecture (affine geometry)">Chern's conjecture (affine geometry)</a> that the <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of a <a href="/wiki/Closed_manifold" title="Closed manifold">compact</a> <a href="/wiki/Affine_manifold" title="Affine manifold">affine manifold</a> vanishes.</li> <li><a href="/wiki/Chern%27s_conjecture_for_hypersurfaces_in_spheres" title="Chern's conjecture for hypersurfaces in spheres">Chern's conjecture for hypersurfaces in spheres</a>, a number of closely related conjectures.</li> <li>Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.<a href="#cite_note-51">[51]</a></li> <li>The <a href="/wiki/Filling_area_conjecture" title="Filling area conjecture">filling area conjecture</a>, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length<a href="#cite_note-52">[52]</a></li> <li>The <a href="/wiki/Hopf_conjecture" title="Hopf conjecture">Hopf conjectures</a> relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds<a href="#cite_note-53">[53]</a></li> <li><a href="/wiki/Yau%27s_conjecture_on_the_first_eigenvalue" title="Yau's conjecture on the first eigenvalue">Yau's conjecture on the first eigenvalue</a> that the first <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalue</a> for the <a href="/wiki/Laplace%E2%80%93Beltrami_operator" title="Laplace–Beltrami operator">Laplace–Beltrami operator</a> on an embedded <a href="/wiki/Minimal_surface" title="Minimal surface">minimal hypersurface</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d94abac312d4d24606e0465b5468606a3923068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n+1}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Discrete_geometry">Discrete geometry</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kissing-3d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Kissing-3d.png/220px-Kissing-3d.png" decoding="async" width="220" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Kissing-3d.png/330px-Kissing-3d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8f/Kissing-3d.png 2x" data-file-width="424" data-file-height="437" /></a><figcaption>In three dimensions, the <a href="/wiki/Kissing_number_problem" class="mw-redirect" title="Kissing number problem">kissing number</a> is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a>.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.</figcaption></figure> <ul><li>The <a href="/wiki/Big-line-big-clique_conjecture" title="Big-line-big-clique conjecture">big-line-big-clique conjecture</a> on the existence of either many collinear points or many mutually visible points in large planar point sets<a href="#cite_note-54">[54]</a></li> <li>The <a href="/wiki/Hadwiger_conjecture_(combinatorial_geometry)" title="Hadwiger conjecture (combinatorial geometry)">Hadwiger conjecture</a> on covering n-dimensional convex bodies with at most 2n smaller copies<a href="#cite_note-55">[55]</a></li> <li>Solving the <a href="/wiki/Happy_ending_problem" title="Happy ending problem">happy ending problem</a> for arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"><a href="#cite_note-56">[56]</a></li> <li>Improving lower and upper bounds for the <a href="/wiki/Heilbronn_triangle_problem" title="Heilbronn triangle problem">Heilbronn triangle problem</a>.</li> <li><a href="/wiki/Kalai%27s_3%5Ed_conjecture" title="Kalai's 3^d conjecture">Kalai's 3d conjecture</a> on the least possible number of faces of <a href="/wiki/Point_symmetry" class="mw-redirect" title="Point symmetry">centrally symmetric</a> <a href="/wiki/Polytopes" class="mw-redirect" title="Polytopes">polytopes</a>.<a href="#cite_note-kalai-57">[57]</a></li> <li>The <a href="/wiki/Kobon_triangle_problem" title="Kobon triangle problem">Kobon triangle problem</a> on triangles in line arrangements<a href="#cite_note-58">[58]</a></li> <li>The <a href="/wiki/Kusner_conjecture" class="mw-redirect" title="Kusner conjecture">Kusner conjecture</a>: at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8106478cb4da6af49992eeb3a3b8690d27797ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.176ex;" alt="{\displaystyle 2d}"> points can be equidistant in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"> spaces<a href="#cite_note-59">[59]</a></li> <li>The <a href="/wiki/McMullen_problem" title="McMullen problem">McMullen problem</a> on projectively transforming sets of points into <a href="/wiki/Convex_position" title="Convex position">convex position</a><a href="#cite_note-60">[60]</a></li> <li><a href="/wiki/Opaque_forest_problem" class="mw-redirect" title="Opaque forest problem">Opaque forest problem</a> on finding <a href="/wiki/Opaque_set" title="Opaque set">opaque sets</a> for various planar shapes</li> <li><a href="/wiki/Unit_distance_graph#Counting_unit_distances" title="Unit distance graph">How many unit distances</a> can be determined by a set of n points in the Euclidean plane?<a href="#cite_note-61">[61]</a></li> <li>Finding matching upper and lower bounds for <a href="/wiki/K-set_(geometry)" title="K-set (geometry)">k-sets</a> and halving lines<a href="#cite_note-62">[62]</a></li> <li><a href="/wiki/Tripod_packing" title="Tripod packing">Tripod packing</a>:<a href="#cite_note-63">[63]</a> how many tripods can have their apexes packed into a given cube?</li></ul> <div class="mw-heading mw-heading4"><h4 id="Euclidean_geometry">Euclidean geometry</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></div> <ul><li>The <a href="/wiki/Atiyah_conjecture_on_configurations" title="Atiyah conjecture on configurations">Atiyah conjecture on configurations</a> on the invertibility of a certain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-by-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> matrix depending on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"><a href="#cite_note-64">[64]</a></li> <li><a href="/wiki/Bellman%27s_lost-in-a-forest_problem" title="Bellman's lost-in-a-forest problem">Bellman's lost-in-a-forest problem</a> – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation<a href="#cite_note-65">[65]</a></li> <li><a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?<a href="#cite_note-66">[66]</a></li> <li>Danzer's problem and Conway's dead fly problem – do <a href="/wiki/Danzer_set" title="Danzer set">Danzer sets</a> of bounded density or bounded separation exist?<a href="#cite_note-67">[67]</a></li> <li><a href="/wiki/Dissection_into_orthoschemes" title="Dissection into orthoschemes">Dissection into orthoschemes</a> – is it possible for <a href="/wiki/Simplex" title="Simplex">simplices</a> of every dimension?<a href="#cite_note-68">[68]</a></li> <li><a href="/wiki/Ehrhart%27s_volume_conjecture" title="Ehrhart's volume conjecture">Ehrhart's volume conjecture</a>: a convex body <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> dimensions containing a single lattice point in its interior as its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> cannot have volume greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)^{n}/n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)^{n}/n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067d8ddbf91c7487fb330033be927c283308df57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.629ex; height:2.843ex;" alt="{\displaystyle (n+1)^{n}/n!}"></li> <li><a href="/wiki/Falconer%27s_conjecture" title="Falconer's conjecture">Falconer's conjecture</a>: sets of Hausdorff dimension greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582b6455b1ff5f4fb027024a8b1458687dc8ed74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.541ex; height:2.843ex;" alt="{\displaystyle d/2}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"> must have a distance set of nonzero <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a><a href="#cite_note-69">[69]</a></li> <li>The values of the <a href="/wiki/Hermite_constant" title="Hermite constant">Hermite constants</a> for dimensions other than 1–8 and 24</li> <li><a href="/wiki/Inscribed_square_problem" title="Inscribed square problem">Inscribed square problem</a>, also known as <a href="/wiki/Toeplitz%27_conjecture" class="mw-redirect" title="Toeplitz' conjecture">Toeplitz' conjecture</a> and the square peg problem – does every <a href="/wiki/Jordan_curve" class="mw-redirect" title="Jordan curve">Jordan curve</a> have an inscribed square?<a href="#cite_note-matschke-70">[70]</a></li> <li>The <a href="/wiki/Kakeya_conjecture" class="mw-redirect" title="Kakeya conjecture">Kakeya conjecture</a> – do <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional sets that contain a unit line segment in every direction necessarily have <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a> and <a href="/wiki/Minkowski_dimension" class="mw-redirect" title="Minkowski dimension">Minkowski dimension</a> equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">?<a href="#cite_note-71">[71]</a></li> <li>The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the <a href="/wiki/Weaire%E2%80%93Phelan_structure" title="Weaire–Phelan structure">Weaire–Phelan structure</a> as a solution to the Kelvin problem<a href="#cite_note-72">[72]</a></li> <li><a href="/wiki/Lebesgue%27s_universal_covering_problem" title="Lebesgue's universal covering problem">Lebesgue's universal covering problem</a> on the minimum-area convex shape in the plane that can cover any shape of diameter one<a href="#cite_note-73">[73]</a></li> <li><a href="/wiki/Mahler_volume" title="Mahler volume">Mahler's conjecture</a> on the product of the volumes of a <a href="/wiki/Central_symmetry" class="mw-redirect" title="Central symmetry">centrally symmetric</a> <a href="/wiki/Convex_body" title="Convex body">convex body</a> and its <a href="/wiki/Polar_set" title="Polar set">polar</a>.<a href="#cite_note-74">[74]</a></li> <li><a href="/wiki/Moser%27s_worm_problem" title="Moser's worm problem">Moser's worm problem</a> – what is the smallest area of a shape that can cover every unit-length curve in the plane?<a href="#cite_note-75">[75]</a></li> <li>The <a href="/wiki/Moving_sofa_problem" title="Moving sofa problem">moving sofa problem</a> – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?<a href="#cite_note-76">[76]</a></li> <li>Does every convex polyhedron have <a href="/wiki/Prince_Rupert%27s_cube#Generalizations" title="Prince Rupert's cube">Rupert's property</a>?<a href="#cite_note-cyz-77">[77]</a><a href="#cite_note-styu-78">[78]</a></li> <li><a href="/wiki/Shephard%27s_conjecture" class="mw-redirect" title="Shephard's conjecture">Shephard's problem (a.k.a. Dürer's conjecture)</a> – does every <a href="/wiki/Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">convex polyhedron</a> have a <a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">net</a>, or simple edge-unfolding?<a href="#cite_note-79">[79]</a><a href="#cite_note-80">[80]</a></li> <li>Is there a non-convex polyhedron without self-intersections with <a href="/wiki/Szilassi_polyhedron" title="Szilassi polyhedron">more than seven faces</a>, all of which share an edge with each other?</li> <li>The <a href="/wiki/Thomson_problem" title="Thomson problem">Thomson problem</a> – what is the minimum energy configuration of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> mutually-repelling particles on a unit sphere?<a href="#cite_note-81">[81]</a></li> <li>Convex <a href="/wiki/Uniform_5-polytope" title="Uniform 5-polytope">uniform 5-polytopes</a> – find and classify the complete set of these shapes<a href="#cite_note-82">[82]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graph_theory">Graph theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></div> <div class="mw-heading mw-heading4"><h4 id="Algebraic_graph_theory">Algebraic graph theory</h4></div> <ul><li><a href="/wiki/Babai%27s_problem" title="Babai's problem">Babai's problem</a>: which groups are Babai invariant groups?</li> <li><a href="/wiki/Brouwer%27s_conjecture" title="Brouwer's conjecture">Brouwer's conjecture</a> on upper bounds for sums of <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> of <a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacians</a> of graphs in terms of their number of edges</li></ul> <div class="mw-heading mw-heading4"><h4 id="Games_on_graphs">Games on graphs</h4></div> <ul><li><a href="/wiki/Graham%27s_pebbling_conjecture" class="mw-redirect" title="Graham's pebbling conjecture">Graham's pebbling conjecture</a> on the pebbling number of Cartesian products of graphs<a href="#cite_note-83">[83]</a></li> <li>Meyniel's conjecture that <a href="/wiki/Cop_number" title="Cop number">cop number</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({\sqrt {n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\sqrt {n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5526ab1252c0f682bbe07c0ad67c0f29de5522b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.913ex; height:3.009ex;" alt="{\displaystyle O({\sqrt {n}})}"><a href="#cite_note-84">[84]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Graph_coloring_and_labeling">Graph coloring and labeling</h4></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg/220px-Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg.png" decoding="async" width="220" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg/330px-Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg/440px-Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture.svg.png 2x" data-file-width="574" data-file-height="612" /></a><figcaption>An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.</figcaption></figure> <ul><li>The <a href="/wiki/Graph_factorization#1-factorization_conjecture" title="Graph factorization">1-factorization conjecture</a> that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is odd or even and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq n,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq n,n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891a20f063981733ec4207e0edd4fe4cdab2fb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.136ex; height:2.509ex;" alt="{\displaystyle k\geq n,n-1}"> respectively, then a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-<a href="/wiki/Regular_graph" title="Regular graph">regular graph</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}"> vertices is <a href="/wiki/Graph_factorization#1-factorization" title="Graph factorization">1-factorable</a>. <ul><li>The <a href="/wiki/Graph_factorization#Perfect_1-factorization" title="Graph factorization">perfect 1-factorization conjecture</a> that every <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a> on an even number of vertices admits a <a href="/wiki/Graph_factorization#Perfect_1-factorization" title="Graph factorization">perfect 1-factorization</a>.</li></ul></li> <li><a href="/wiki/Cereceda%27s_conjecture" title="Cereceda's conjecture">Cereceda's conjecture</a> on the diameter of the space of colorings of degenerate graphs<a href="#cite_note-85">[85]</a></li> <li>The <a href="/wiki/Earth%E2%80%93Moon_problem" title="Earth–Moon problem">Earth–Moon problem</a>: what is the maximum chromatic number of biplanar graphs?<a href="#cite_note-86">[86]</a></li> <li>The <a href="/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture" title="Erdős–Faber–Lovász conjecture">Erdős–Faber–Lovász conjecture</a> on coloring unions of cliques<a href="#cite_note-87">[87]</a></li> <li>The <a href="/wiki/Graceful_labeling" title="Graceful labeling">graceful tree conjecture</a> that every tree admits a graceful labeling <ul><li><a href="/wiki/Graceful_labeling" title="Graceful labeling">Rosa's conjecture</a> that all <a href="/wiki/Cactus_graph#Triangular_cactus" title="Cactus graph">triangular cacti</a> are graceful or nearly-graceful</li></ul></li> <li>The <a href="/wiki/Gy%C3%A1rf%C3%A1s%E2%80%93Sumner_conjecture" title="Gyárfás–Sumner conjecture">Gyárfás–Sumner conjecture</a> on χ-boundedness of graphs with a forbidden induced tree<a href="#cite_note-88">[88]</a></li> <li>The <a href="/wiki/Hadwiger_conjecture_(graph_theory)" title="Hadwiger conjecture (graph theory)">Hadwiger conjecture</a> relating coloring to clique minors<a href="#cite_note-89">[89]</a></li> <li>The <a href="/wiki/Hadwiger%E2%80%93Nelson_problem" title="Hadwiger–Nelson problem">Hadwiger–Nelson problem</a> on the chromatic number of unit distance graphs<a href="#cite_note-90">[90]</a></li> <li><a href="/wiki/Petersen_graph#Petersen_coloring_conjecture" title="Petersen graph">Jaeger's Petersen-coloring conjecture</a>: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph<a href="#cite_note-91">[91]</a></li> <li>The <a href="/wiki/List_coloring_conjecture" class="mw-redirect" title="List coloring conjecture">list coloring conjecture</a>: for every graph, the list chromatic index equals the chromatic index<a href="#cite_note-92">[92]</a></li> <li>The <a href="/wiki/Overfull_graph#Overfull_conjecture" title="Overfull graph">overfull conjecture</a> that a graph with maximum degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (G)\geq n/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (G)\geq n/3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c19772d97806e15e88aeeee446dc54185a0ec2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.39ex; height:2.843ex;" alt="{\displaystyle \Delta (G)\geq n/3}"> is <a href="/wiki/Vizing%27s_theorem" title="Vizing's theorem">class 2</a> if and only if it has an <a href="/wiki/Overfull_graph" title="Overfull graph">overfull subgraph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (S)=\Delta (G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (S)=\Delta (G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d45cf9c11bc5b7296673226c931d7f209e6f293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.915ex; height:2.843ex;" alt="{\displaystyle \Delta (S)=\Delta (G)}">.</li> <li>The <a href="/wiki/Total_coloring_conjecture" class="mw-redirect" title="Total coloring conjecture">total coloring conjecture</a> of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree<a href="#cite_note-93">[93]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Graph_drawing_and_embedding">Graph drawing and embedding</h4></div> <ul><li>The <a href="/wiki/Albertson_conjecture" title="Albertson conjecture">Albertson conjecture</a>: the crossing number can be lower-bounded by the crossing number of a <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a> with the same <a href="/wiki/Chromatic_number" class="mw-redirect" title="Chromatic number">chromatic number</a><a href="#cite_note-94">[94]</a></li> <li><a href="/wiki/Conway%27s_thrackle_conjecture" class="mw-redirect" title="Conway's thrackle conjecture">Conway's thrackle conjecture</a><a href="#cite_note-95">[95]</a> that <a href="/wiki/Thrackle" title="Thrackle">thrackles</a> cannot have more edges than vertices</li> <li>The <a href="/wiki/GNRS_conjecture" title="GNRS conjecture">GNRS conjecture</a> on whether minor-closed graph families have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361ddd720474aa41cb05453e03424fb7999d3b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{1}}"> embeddings with bounded distortion<a href="#cite_note-96">[96]</a></li> <li><a href="/wiki/Harborth%27s_conjecture" title="Harborth's conjecture">Harborth's conjecture</a>: every planar graph can be drawn with integer edge lengths<a href="#cite_note-97">[97]</a></li> <li><a href="/wiki/Negami%27s_conjecture" class="mw-redirect" title="Negami's conjecture">Negami's conjecture</a> on projective-plane embeddings of graphs with planar covers<a href="#cite_note-98">[98]</a></li> <li>The <a href="/wiki/Greedy_embedding#Planar_graphs" title="Greedy embedding">strong Papadimitriou–Ratajczak conjecture</a>: every polyhedral graph has a convex greedy embedding<a href="#cite_note-99">[99]</a></li> <li><a href="/wiki/Tur%C3%A1n%27s_brick_factory_problem" title="Turán's brick factory problem">Turán's brick factory problem</a> – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?<a href="#cite_note-100">[100]</a></li></ul> <ul><li><a href="/wiki/Universal_point_set" title="Universal point set">Universal point sets</a> of subquadratic size for planar graphs<a href="#cite_note-101">[101]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Restriction_of_graph_parameters">Restriction of graph parameters</h4></div> <ul><li><a href="/wiki/Conway%27s_99-graph_problem" title="Conway's 99-graph problem">Conway's 99-graph problem</a>: does there exist a <a href="/wiki/Strongly_regular_graph" title="Strongly regular graph">strongly regular graph</a> with parameters (99,14,1,2)?<a href="#cite_note-102">[102]</a></li> <li><a href="/wiki/Degree_diameter_problem" title="Degree diameter problem">Degree diameter problem</a>: given two positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d,k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb3254902512cb166d72568bbcd0b1c68cd0c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.461ex; height:2.509ex;" alt="{\displaystyle d,k}">, what is the largest graph of diameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> such that all vertices have degrees at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">?</li> <li>Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an <a href="/wiki/Apex_graph" title="Apex graph">apex graph</a><a href="#cite_note-103">[103]</a></li> <li>Does a <a href="/wiki/Moore_graph" title="Moore graph">Moore graph</a> with girth 5 and degree 57 exist?<a href="#cite_note-104">[104]</a></li> <li>Do there exist infinitely many <a href="/wiki/Strongly_regular_graph" title="Strongly regular graph">strongly regular</a> <a href="/wiki/Geodetic_graph" title="Geodetic graph">geodetic graphs</a>, or any strongly regular geodetic graphs that are not Moore graphs?<a href="#cite_note-105">[105]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Subgraphs">Subgraphs</h4></div> <ul><li><a href="/wiki/Barnette%27s_conjecture" title="Barnette's conjecture">Barnette's conjecture</a>: every cubic bipartite three-connected planar graph has a Hamiltonian cycle<a href="#cite_note-106">[106]</a></li> <li><a href="/wiki/Gilbert%E2%80%93Pollack_conjecture_on_the_Steiner_ratio_of_the_Euclidean_plane" class="mw-redirect" title="Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane">Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane</a> that the Steiner ratio is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3245e1141ec36a954dd702c886bba16d8c6cb057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3}}/2}"></li> <li><a href="/wiki/Graph_toughness" title="Graph toughness">Chvátal's toughness conjecture</a>, that there is a number t such that every t-tough graph is Hamiltonian<a href="#cite_note-107">[107]</a></li> <li>The <a href="/wiki/Cycle_double_cover_conjecture" class="mw-redirect" title="Cycle double cover conjecture">cycle double cover conjecture</a>: every bridgeless graph has a family of cycles that includes each edge twice<a href="#cite_note-108">[108]</a></li> <li>The <a href="/wiki/Erd%C5%91s%E2%80%93Gy%C3%A1rf%C3%A1s_conjecture" title="Erdős–Gyárfás conjecture">Erdős–Gyárfás conjecture</a> on cycles with power-of-two lengths in cubic graphs<a href="#cite_note-109">[109]</a></li> <li>The <a href="/wiki/Erd%C5%91s%E2%80%93Hajnal_conjecture" title="Erdős–Hajnal conjecture">Erdős–Hajnal conjecture</a> on large cliques or independent sets in graphs with a forbidden induced subgraph<a href="#cite_note-110">[110]</a></li> <li>The <a href="/wiki/Linear_arboricity" title="Linear arboricity">linear arboricity</a> conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree<a href="#cite_note-111">[111]</a></li> <li>The <a href="/wiki/Lov%C3%A1sz_conjecture" title="Lovász conjecture">Lovász conjecture</a> on Hamiltonian paths in symmetric graphs<a href="#cite_note-112">[112]</a></li> <li>The <a href="/wiki/Oberwolfach_problem" title="Oberwolfach problem">Oberwolfach problem</a> on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.<a href="#cite_note-113">[113]</a></li> <li>What is the largest possible <a href="/wiki/Pathwidth" title="Pathwidth">pathwidth</a> of an n-vertex <a href="/wiki/Cubic_graph" title="Cubic graph">cubic graph</a>?<a href="#cite_note-114">[114]</a></li> <li>The <a href="/wiki/Reconstruction_conjecture" title="Reconstruction conjecture">reconstruction conjecture</a> and <a href="/wiki/New_digraph_reconstruction_conjecture" title="New digraph reconstruction conjecture">new digraph reconstruction conjecture</a> on whether a graph is uniquely determined by its vertex-deleted subgraphs.<a href="#cite_note-115">[115]</a><a href="#cite_note-116">[116]</a></li> <li>The <a href="/wiki/Snake-in-the-box" title="Snake-in-the-box">snake-in-the-box</a> problem: what is the longest possible <a href="/wiki/Induced_path" title="Induced path">induced path</a> in an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional <a href="/wiki/Hypercube" title="Hypercube">hypercube</a> graph?</li> <li><a href="/wiki/Sumner%27s_conjecture" title="Sumner's conjecture">Sumner's conjecture</a>: does every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670a78b7547a73a8894aa40bc55ddea996fd38bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.369ex; height:2.843ex;" alt="{\displaystyle (2n-2)}">-vertex tournament contain as a subgraph every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-vertex oriented tree?<a href="#cite_note-117">[117]</a></li> <li><a href="/wiki/Szymanski%27s_conjecture" title="Szymanski's conjecture">Szymanski's conjecture</a>: every <a href="/wiki/Permutation" title="Permutation">permutation</a> on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional doubly-<a href="/wiki/Directed_graph" title="Directed graph">directed</a> <a href="/wiki/Hypercube_graph" title="Hypercube graph">hypercube graph</a> can be routed with edge-disjoint <a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">paths</a>.</li> <li><a href="/wiki/Tuza%27s_conjecture" title="Tuza's conjecture">Tuza's conjecture</a>: if the maximum number of disjoint triangles is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }">, can all triangles be hit by a set of at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2654d0632cb4b4cd30cfddb80a75ba7d743216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.395ex; height:2.176ex;" alt="{\displaystyle 2\nu }"> edges?<a href="#cite_note-118">[118]</a></li> <li><a href="/wiki/Vizing%27s_conjecture" title="Vizing's conjecture">Vizing's conjecture</a> on the <a href="/wiki/Domination_number" class="mw-redirect" title="Domination number">domination number</a> of <a href="/wiki/Cartesian_product_of_graphs" title="Cartesian product of graphs">cartesian products of graphs</a><a href="#cite_note-119">[119]</a></li> <li><a href="/wiki/Zarankiewicz_problem" title="Zarankiewicz problem">Zarankiewicz problem</a>: how many edges can there be in a <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graph</a> on a given number of vertices with no <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite subgraphs</a> of a given size?</li></ul> <div class="mw-heading mw-heading4"><h4 id="Word-representation_of_graphs">Word-representation of graphs</h4></div> <ul><li>Are there any graphs on n vertices whose <a href="/wiki/Word-representable_graph" title="Word-representable graph">representation</a> requires more than floor(n/2) copies of each letter?<a href="#cite_note-KL15-120">[120]</a><a href="#cite_note-K17-121">[121]</a><a href="#cite_note-KP18-122">[122]</a><a href="#cite_note-KP18-2-123">[123]</a></li> <li>Characterise (non-)<a href="/wiki/Word-representable_graph" title="Word-representable graph">word-representable</a> <a href="/wiki/Planar_graph" title="Planar graph">planar graphs</a><a href="#cite_note-KL15-120">[120]</a><a href="#cite_note-K17-121">[121]</a><a href="#cite_note-KP18-122">[122]</a><a href="#cite_note-KP18-2-123">[123]</a></li> <li>Characterise <a href="/wiki/Word-representable_graph" title="Word-representable graph">word-representable graphs</a> in terms of (induced) forbidden subgraphs.<a href="#cite_note-KL15-120">[120]</a><a href="#cite_note-K17-121">[121]</a><a href="#cite_note-KP18-122">[122]</a><a href="#cite_note-KP18-2-123">[123]</a></li> <li>Characterise <a href="/wiki/Word-representable_graph" title="Word-representable graph">word-representable</a> near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs<a href="#cite_note-Glen2019-124">[124]</a>)</li> <li>Classify graphs with representation number 3, that is, graphs that can be <a href="/wiki/Word-representable_graph" title="Word-representable graph">represented</a> using 3 copies of each letter, but cannot be represented using 2 copies of each letter<a href="#cite_note-Kit2013-3-repr-125">[125]</a></li> <li>Is it true that out of all <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graphs</a>, <a href="/wiki/Crown_graph" title="Crown graph">crown graphs</a> require longest word-representants?<a href="#cite_note-GKP18-126">[126]</a></li> <li>Is the <a href="/wiki/Line_graph" title="Line graph">line graph</a> of a non-<a href="/wiki/Word-representable_graph" title="Word-representable graph">word-representable</a> graph always non-<a href="/wiki/Word-representable_graph" title="Word-representable graph">word-representable</a>?<a href="#cite_note-KL15-120">[120]</a><a href="#cite_note-K17-121">[121]</a><a href="#cite_note-KP18-122">[122]</a><a href="#cite_note-KP18-2-123">[123]</a></li> <li>Which (hard) problems on graphs can be translated to words <a href="/wiki/Word-representable_graph" title="Word-representable graph">representing</a> them and solved on words (efficiently)?<a href="#cite_note-KL15-120">[120]</a><a href="#cite_note-K17-121">[121]</a><a href="#cite_note-KP18-122">[122]</a><a href="#cite_note-KP18-2-123">[123]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Miscellaneous_graph_theory">Miscellaneous graph theory</h4></div> <ul><li>The <a href="/wiki/Implicit_graph_conjecture" class="mw-redirect" title="Implicit graph conjecture">implicit graph conjecture</a> on the existence of implicit representations for slowly-growing <a href="/wiki/Hereditary_property#In_graph_theory" title="Hereditary property">hereditary families of graphs</a><a href="#cite_note-127">[127]</a></li> <li><a href="/wiki/Ryser%27s_conjecture" title="Ryser's conjecture">Ryser's conjecture</a> relating the maximum <a href="/wiki/Matching_in_hypergraphs" title="Matching in hypergraphs">matching</a> size and minimum <a href="/wiki/Vertex_cover_in_hypergraphs" title="Vertex cover in hypergraphs">transversal</a> size in <a href="/wiki/Hypergraph" title="Hypergraph">hypergraphs</a></li> <li>The <a href="/wiki/Second_neighborhood_problem" title="Second neighborhood problem">second neighborhood problem</a>: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?<a href="#cite_note-128">[128]</a></li> <li><a href="/wiki/Sidorenko%27s_conjecture" title="Sidorenko's conjecture">Sidorenko's conjecture</a> on <a href="/wiki/Homomorphism_density" title="Homomorphism density">homomorphism densities</a> of graphs in <a href="/wiki/Graphon" title="Graphon">graphons</a></li> <li>Tutte's conjectures: <ul><li>every bridgeless graph has a <a href="/wiki/Nowhere-zero_flows" class="mw-redirect" title="Nowhere-zero flows">nowhere-zero 5-flow</a><a href="#cite_note-129">[129]</a></li> <li>every <a href="/wiki/Petersen_graph" title="Petersen graph">Petersen</a>-<a href="/wiki/Graph_minor" title="Graph minor">minor</a>-free bridgeless graph has a nowhere-zero 4-flow<a href="#cite_note-130">[130]</a></li></ul></li> <li><a href="/wiki/Woodall%27s_conjecture" title="Woodall's conjecture">Woodall's conjecture</a> that the minimum number of edges in a <a href="/wiki/Dicut" title="Dicut">dicut</a> of a <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> is equal to the maximum number of disjoint <a href="/wiki/Dijoin" title="Dijoin">dijoins</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Model_theory_and_formal_languages">Model theory and formal languages</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Model_theory" title="Model theory">Model theory</a> and <a href="/wiki/Formal_languages" class="mw-redirect" title="Formal languages">formal languages</a></div> <ul><li>The <a href="/wiki/Stable_group" title="Stable group">Cherlin–Zilber conjecture</a>: A simple group whose first-order theory is <a href="/wiki/Stable_theory" title="Stable theory">stable</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"> is a simple algebraic group over an algebraically closed field.</li> <li><a href="/wiki/Generalized_star_height_problem" class="mw-redirect" title="Generalized star height problem">Generalized star height problem</a>: can all <a href="/wiki/Regular_language" title="Regular language">regular languages</a> be expressed using <a href="/wiki/Regular_expression#Expressive_power_and_compactness" title="Regular expression">generalized regular expressions</a> with limited nesting depths of <a href="/wiki/Kleene_star" title="Kleene star">Kleene stars</a>?</li> <li>For which number fields does <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a> hold?</li> <li>Kueker's conjecture<a href="#cite_note-131">[131]</a></li> <li>The main gap conjecture, e.g. for uncountable <a href="/wiki/First_order_theory" class="mw-redirect" title="First order theory">first order theories</a>, for <a href="/wiki/Abstract_elementary_class" title="Abstract elementary class">AECs</a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}">-saturated models of a countable theory.<a href="#cite_note-:0-132">[132]</a></li> <li>Shelah's categoricity conjecture for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\omega _{1},\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037b84dcf7eb85313f964632c939bce6464a657c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.149ex; height:2.843ex;" alt="{\displaystyle L_{\omega _{1},\omega }}">: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.<a href="#cite_note-:0-132">[132]</a></li> <li>Shelah's eventual categoricity conjecture: For every cardinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> there exists a cardinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cebe02d911e1160383302c9f7f3d493af6dd5d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.566ex; height:2.843ex;" alt="{\displaystyle \mu (\lambda )}"> such that if an <a href="/wiki/Abstract_elementary_class" title="Abstract elementary class">AEC</a> K with LS(K)<= <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> is categorical in a cardinal above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cebe02d911e1160383302c9f7f3d493af6dd5d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.566ex; height:2.843ex;" alt="{\displaystyle \mu (\lambda )}"> then it is categorical in all cardinals above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cebe02d911e1160383302c9f7f3d493af6dd5d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.566ex; height:2.843ex;" alt="{\displaystyle \mu (\lambda )}">.<a href="#cite_note-:0-132">[132]</a><a href="#cite_note-133">[133]</a></li> <li>The stable field conjecture: every infinite field with a <a href="/wiki/Stable_theory" title="Stable theory">stable</a> first-order theory is separably closed.</li> <li>The stable forking conjecture for simple theories<a href="#cite_note-134">[134]</a></li> <li><a href="/wiki/Tarski%27s_exponential_function_problem" title="Tarski's exponential function problem">Tarski's exponential function problem</a>: is the <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">theory</a> of the <a href="/wiki/Real_number" title="Real number">real numbers</a> with the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> <a href="/wiki/Decidability_(logic)#Decidability_of_a_theory" title="Decidability (logic)">decidable</a>?</li> <li>The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?<a href="#cite_note-135">[135]</a></li> <li>The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?<a href="#cite_note-136">[136]</a></li> <li><a href="/wiki/Vaught_conjecture" title="Vaught conjecture">Vaught conjecture</a>: the number of <a href="/wiki/Countable_set" title="Countable set">countable</a> models of a <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> <a href="/wiki/Complete_theory" title="Complete theory">complete theory</a> in a countable <a href="/wiki/Formal_language" title="Formal language">language</a> is either finite, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779da5db4ed54fa334dd92089cdf1c284e45febb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.231ex; height:2.676ex;" alt="{\displaystyle 2^{\aleph _{0}}}">.</li></ul> <ul><li>Assume K is the class of models of a countable first order theory omitting countably many <a href="/wiki/Type_(model_theory)" title="Type (model theory)">types</a>. If K has a model of cardinality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{\omega _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\omega _{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5581018c2ccc2eab23589bcc75998188cc9556b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.506ex; height:2.843ex;" alt="{\displaystyle \aleph _{\omega _{1}}}"> does it have a model of cardinality continuum?<a href="#cite_note-137">[137]</a></li> <li>Do the <a href="/wiki/Henson_graph" title="Henson graph">Henson graphs</a> have the <a href="/wiki/Finite_model_property" title="Finite model property">finite model property</a>?</li> <li>Does a finitely presented homogeneous structure for a finite relational language have finitely many <a href="/wiki/Reduct" title="Reduct">reducts</a>?</li> <li>Does there exist an <a href="/wiki/O-minimal" class="mw-redirect" title="O-minimal">o-minimal</a> first order theory with a trans-exponential (rapid growth) function?</li> <li>If the class of atomic models of a complete first order theory is <a href="/wiki/Categorical_(model_theory)" class="mw-redirect" title="Categorical (model theory)">categorical</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008ec35e700beeb6f6ac979c7ed7bfec35f85725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle \aleph _{n}}">, is it categorical in every cardinal?<a href="#cite_note-138">[138]</a><a href="#cite_note-139">[139]</a></li> <li>Is every infinite, minimal field of characteristic zero <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)</li> <li>Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?<a href="#cite_note-140">[140]</a></li> <li>Is the theory of the field of Laurent series over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"> <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>? of the field of polynomials over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">?</li> <li>Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?<a href="#cite_note-141">[141]</a></li></ul> <ul><li>Determine the structure of Keisler's order.<a href="#cite_note-142">[142]</a><a href="#cite_note-143">[143]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Probability_theory">Probability theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></div> <ul><li><a href="/wiki/Ibragimov%E2%80%93Iosifescu_conjecture_for_%CF%86-mixing_sequences" title="Ibragimov–Iosifescu conjecture for φ-mixing sequences">Ibragimov–Iosifescu conjecture for φ-mixing sequences</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main page: <a href="/wiki/Category:Unsolved_problems_in_number_theory" title="Category:Unsolved problems in number theory">Category:Unsolved problems in number theory</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Number_theory" title="Number theory">Number theory</a></div> <div class="mw-heading mw-heading4"><h4 id="General">General</h4></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Perfect_number_Cuisenaire_rods_6_exact.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Perfect_number_Cuisenaire_rods_6_exact.svg/220px-Perfect_number_Cuisenaire_rods_6_exact.svg.png" decoding="async" width="220" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Perfect_number_Cuisenaire_rods_6_exact.svg/330px-Perfect_number_Cuisenaire_rods_6_exact.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Perfect_number_Cuisenaire_rods_6_exact.svg/440px-Perfect_number_Cuisenaire_rods_6_exact.svg.png 2x" data-file-width="437" data-file-height="345" /></a><figcaption>6 is a <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a> because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.</figcaption></figure> <ul><li><a href="/wiki/Special_values_of_L-functions" title="Special values of L-functions">Beilinson's conjectures</a></li> <li><a href="/wiki/Brocard%27s_problem" title="Brocard's problem">Brocard's problem</a>: are there any integer solutions to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!+1=m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!+1=m^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3aa826f4474ccacd153c6cdc10759f3e515884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.238ex; height:2.843ex;" alt="{\displaystyle n!+1=m^{2}}"> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4,5,7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=4,5,7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd770c28a5c336cbc5bcd85551c181ffa0a7a009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.048ex; height:2.509ex;" alt="{\displaystyle n=4,5,7}">?</li> <li><a href="/wiki/B%C3%BCchi%27s_problem" title="Büchi's problem">Büchi's problem</a> on sufficiently large sequences of square numbers with constant second difference.</li> <li><a href="/wiki/Carmichael%27s_totient_function_conjecture" title="Carmichael's totient function conjecture">Carmichael's totient function conjecture</a>: do all values of <a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a> have <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a> greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">?</li> <li><a href="/wiki/Casas-Alvero_conjecture" title="Casas-Alvero conjecture">Casas-Alvero conjecture</a>: if a polynomial of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> defined over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> has a factor in common with its first through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0195b64ba44bcc80b4c98e9d34256b4043fe519e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.219ex; height:2.343ex;" alt="{\displaystyle d-1}">-th derivative, then must <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> be the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">-th power of a linear polynomial?</li> <li><a href="/wiki/Aliquot_sequence#Catalan-Dickson_conjecture" title="Aliquot sequence">Catalan–Dickson conjecture on aliquot sequences</a>: no <a href="/wiki/Aliquot_sequence" title="Aliquot sequence">aliquot sequences</a> are infinite but non-repeating.</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Ulam_problem" title="Erdős–Ulam problem">Erdős–Ulam problem</a>: is there a <a href="/wiki/Dense_set" title="Dense set">dense set</a> of points in the plane all at rational distances from one-another?</li> <li><a href="/wiki/Van_der_Corput%27s_method#Exponent_pairs" title="Van der Corput's method">Exponent pair conjecture</a>: for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}">, is the pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\epsilon ,1/2+\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\epsilon ,1/2+\epsilon )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520a6878e45c22c247a7f6f4b1ee05e3fa1ca828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.059ex; height:2.843ex;" alt="{\displaystyle (\epsilon ,1/2+\epsilon )}"> an <a href="/wiki/Van_der_Corput%27s_method#Exponent_pairs" title="Van der Corput's method">exponent pair</a>?</li> <li>The <a href="/wiki/Gauss_circle_problem" title="Gauss circle problem">Gauss circle problem</a>: how far can the number of integer points in a circle centered at the origin be from the area of the circle?</li> <li><a href="/wiki/Grand_Riemann_hypothesis" title="Grand Riemann hypothesis">Grand Riemann hypothesis</a>: do the nontrivial zeros of all <a href="/wiki/Automorphic_L-function" title="Automorphic L-function">automorphic L-functions</a> lie on the critical line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2+it}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2+it}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2df2af523085c4237a3b99695f1063ac8412776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.97ex; height:2.843ex;" alt="{\displaystyle 1/2+it}"> with real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">? <ul><li><a href="/wiki/Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a>: do the nontrivial zeros of all <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a> lie on the critical line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2+it}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2+it}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2df2af523085c4237a3b99695f1063ac8412776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.97ex; height:2.843ex;" alt="{\displaystyle 1/2+it}"> with real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">? <ul><li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>: do the nontrivial zeros of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> lie on the critical line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2+it}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2+it}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2df2af523085c4237a3b99695f1063ac8412776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.97ex; height:2.843ex;" alt="{\displaystyle 1/2+it}"> with real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">?</li></ul></li></ul></li> <li><a href="/wiki/Grimm%27s_conjecture" title="Grimm's conjecture">Grimm's conjecture</a>: each element of a set of consecutive <a href="/wiki/Composite_number" title="Composite number">composite numbers</a> can be assigned a distinct <a href="/wiki/Prime_number" title="Prime number">prime number</a> that divides it.</li> <li><a href="/wiki/Hall%27s_conjecture" title="Hall's conjecture">Hall's conjecture</a>: for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}">, there is some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\epsilon )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6d84254c3840f120614be54230e5b80e759359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.76ex; height:2.843ex;" alt="{\displaystyle c(\epsilon )}"> such that either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542b1833d59c3ed167c061e27c140e08dde91425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.697ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y^{2}-x^{3}|>c(\epsilon )x^{1/2-\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>ϵ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |y^{2}-x^{3}|>c(\epsilon )x^{1/2-\epsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/787b125a2be82ce3368461af57452862d90df916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.565ex; height:3.343ex;" alt="{\displaystyle |y^{2}-x^{3}|>c(\epsilon )x^{1/2-\epsilon }}">.</li> <li><a href="/wiki/Hardy%E2%80%93Littlewood_zeta_function_conjectures" title="Hardy–Littlewood zeta function conjectures">Hardy–Littlewood zeta function conjectures</a></li> <li><a href="/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture" title="Hilbert–Pólya conjecture">Hilbert–Pólya conjecture</a>: the nontrivial zeros of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> correspond to <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> of a <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a>.</li> <li><a href="/wiki/Hilbert%27s_eleventh_problem" title="Hilbert's eleventh problem">Hilbert's eleventh problem</a>: classify <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> over <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a>.</li> <li><a href="/wiki/Hilbert%27s_ninth_problem" title="Hilbert's ninth problem">Hilbert's ninth problem</a>: find the most general <a href="/wiki/Reciprocity_law" title="Reciprocity law">reciprocity law</a> for the <a href="/wiki/Hilbert_symbol" title="Hilbert symbol">norm residues</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-th order in a general <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number field</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is a power of a prime.</li> <li><a href="/wiki/Hilbert%27s_twelfth_problem" title="Hilbert's twelfth problem">Hilbert's twelfth problem</a>: extend the <a href="/wiki/Kronecker%E2%80%93Weber_theorem" title="Kronecker–Weber theorem">Kronecker–Weber theorem</a> on <a href="/wiki/Abelian_extension" title="Abelian extension">Abelian extensions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> to any base number field.</li> <li>Keating–Snaith conjecture concerning the asymptotics of an integral involving the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a><a href="#cite_note-144">[144]</a></li> <li><a href="/wiki/Lehmer%27s_totient_problem" title="Lehmer's totient problem">Lehmer's totient problem</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb3dbe542ca7d51c4f32e32cefb8572edb26ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.589ex; height:2.843ex;" alt="{\displaystyle \phi (n)}"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}">, must <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> be prime?</li> <li><a href="/wiki/Leopoldt%27s_conjecture" title="Leopoldt's conjecture">Leopoldt's conjecture</a>: a <a href="/wiki/P-adic_number" title="P-adic number">p-adic</a> analogue of the <a href="/wiki/Dirichlet%27s_unit_theorem#The_regulator" title="Dirichlet's unit theorem">regulator</a> of an <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number field</a> does not vanish.</li> <li><a href="/wiki/Lindel%C3%B6f_hypothesis" title="Lindelöf hypothesis">Lindelöf hypothesis</a> that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (1/2+it)=o(t^{\epsilon })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (1/2+it)=o(t^{\epsilon })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c328e278b5d771fc1010674e8191df940bbb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.649ex; height:2.843ex;" alt="{\displaystyle \zeta (1/2+it)=o(t^{\epsilon })}"> <ul><li>The <a href="/wiki/Bombieri%E2%80%93Vinogradov_theorem" title="Bombieri–Vinogradov theorem">density hypothesis</a> for zeroes of the Riemann zeta function</li></ul></li> <li><a href="/wiki/Littlewood_conjecture" title="Littlewood conjecture">Littlewood conjecture</a>: for any two real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b46b57cfa0011b643037751809904d915c1b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>n</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>n</mi> <mi>α</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>n</mi> <mi>β</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6441b88120c558e3da418676c914aa1a78dd48ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.344ex; height:3.843ex;" alt="{\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vert x\Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Vert x\Vert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55b4869c4465426e7f1efca07aeecf645fc9ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \Vert x\Vert }"> is the distance from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> to the nearest integer.</li> <li><a href="/wiki/Mahler%27s_3/2_problem" title="Mahler's 3/2 problem">Mahler's 3/2 problem</a> that no real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> has the property that the fractional parts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(3/2)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(3/2)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38408a87601f4216211772e392a294add024a43a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.845ex; height:2.843ex;" alt="{\displaystyle x(3/2)^{n}}"> are less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e308a3a46b7fdce07cc09dcab9e8d8f73e37d935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/2}"> for all positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.</li> <li><a href="/wiki/Montgomery%27s_pair_correlation_conjecture" title="Montgomery's pair correlation conjecture">Montgomery's pair correlation conjecture</a>: the normalized pair <a href="/wiki/Correlation_function" title="Correlation function">correlation function</a> between pairs of zeros of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> is the same as the pair correlation function of <a href="/wiki/Random_matrix#Gaussian_ensembles" title="Random matrix">random Hermitian matrices</a>.</li> <li><a href="/wiki/N_conjecture" title="N conjecture">n conjecture</a>: a generalization of the abc conjecture to more than three integers. <ul><li><a href="/wiki/Abc_conjecture" title="Abc conjecture">abc conjecture</a>: for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{rad}}(abc)^{1+\epsilon }<c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>rad</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>ϵ</mi> </mrow> </msup> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{rad}}(abc)^{1+\epsilon }<c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1b89d3c1d5587fcec781653c2205fcb0e712c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.516ex; height:3.176ex;" alt="{\displaystyle {\text{rad}}(abc)^{1+\epsilon }<c}"> is true for only finitely many positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c85465b60dd983e3d20d07b64938bbf40b9220d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.173ex; height:2.343ex;" alt="{\displaystyle a+b=c}">.</li> <li><a href="/wiki/Szpiro%27s_conjecture" title="Szpiro's conjecture">Szpiro's conjecture</a>: for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}">, there is some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\epsilon )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783db856a83f49b75f39c35610143fdb071613f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.52ex; height:2.843ex;" alt="{\displaystyle C(\epsilon )}"> such that, for any elliptic curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> defined over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> with minimal discriminant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"> and conductor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta |\leq C(\epsilon )\cdot f^{6+\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mi>ϵ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta |\leq C(\epsilon )\cdot f^{6+\epsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b43b007fb7b28b4eb50ecf4776cea035216500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.848ex; height:3.176ex;" alt="{\displaystyle |\Delta |\leq C(\epsilon )\cdot f^{6+\epsilon }}">.</li></ul></li> <li><a href="/wiki/Newman%27s_conjecture" title="Newman's conjecture">Newman's conjecture</a>: the <a href="/wiki/Partition_function_(number_theory)" title="Partition function (number theory)">partition function</a> satisfies any arbitrary congruence infinitely often.</li> <li><a href="/wiki/Divisor_summatory_function#Piltz_divisor_problem" title="Divisor summatory function">Piltz divisor problem</a> on bounding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2843b07dce4e8b4c34021e7da821d4cfb05985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.085ex; height:2.843ex;" alt="{\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))}"> <ul><li><a href="/wiki/Divisor_summatory_function#Dirichlet's_divisor_problem" title="Divisor summatory function">Dirichlet's divisor problem</a>: the specific case of the Piltz divisor problem for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></li></ul></li> <li><a href="/wiki/Ramanujan%E2%80%93Petersson_conjecture" title="Ramanujan–Petersson conjecture">Ramanujan–Petersson conjecture</a>: a number of related conjectures that are generalizations of the original conjecture.</li> <li><a href="/wiki/Sato%E2%80%93Tate_conjecture" title="Sato–Tate conjecture">Sato–Tate conjecture</a>: also a number of related conjectures that are generalizations of the original conjecture.</li> <li><a href="/wiki/Scholz_conjecture" title="Scholz conjecture">Scholz conjecture</a>: the length of the shortest <a href="/wiki/Addition_chain" title="Addition chain">addition chain</a> producing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e4bd4ef2f9549d026cbf643a91c0d12a8c6794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}"> is at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> plus the length of the shortest addition chain producing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.</li> <li>Do <a href="/wiki/Siegel_zero" title="Siegel zero">Siegel zeros</a> exist?</li> <li><a href="/wiki/Singmaster%27s_conjecture" title="Singmaster's conjecture">Singmaster's conjecture</a>: is there a finite upper bound on the multiplicities of the entries greater than 1 in <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>?<a href="#cite_note-145">[145]</a></li> <li><a href="/wiki/Vojta%27s_conjecture" title="Vojta's conjecture">Vojta's conjecture</a> on <a href="/wiki/Height_function" title="Height function">heights</a> of points on <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> over <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a>.</li></ul> <ul><li>Are there infinitely many <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a>?</li> <li>Do any <a href="/wiki/Odd_perfect_number" class="mw-redirect" title="Odd perfect number">odd perfect numbers</a> exist?</li> <li>Do <a href="/wiki/Quasiperfect_number" title="Quasiperfect number">quasiperfect numbers</a> exist?</li> <li>Do any non-power of 2 <a href="/wiki/Almost_perfect_number" title="Almost perfect number">almost perfect numbers</a> exist?</li> <li>Are there 65, 66, or 67 <a href="/wiki/Idoneal_number" title="Idoneal number">idoneal numbers</a>?</li> <li>Are there any pairs of <a href="/wiki/Amicable_numbers" title="Amicable numbers">amicable numbers</a> which have opposite parity?</li> <li>Are there any pairs of <a href="/wiki/Betrothed_numbers" title="Betrothed numbers">betrothed numbers</a> which have same parity?</li> <li>Are there any pairs of <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a> <a href="/wiki/Amicable_numbers" title="Amicable numbers">amicable numbers</a>?</li> <li>Are there infinitely many <a href="/wiki/Amicable_numbers" title="Amicable numbers">amicable numbers</a>?</li> <li>Are there infinitely many <a href="/wiki/Betrothed_numbers" title="Betrothed numbers">betrothed numbers</a>?</li> <li>Are there infinitely many <a href="/wiki/Giuga_number" title="Giuga number">Giuga numbers</a>?</li> <li>Does every <a href="/wiki/Rational_number" title="Rational number">rational number</a> with an odd denominator have an <a href="/wiki/Odd_greedy_expansion" title="Odd greedy expansion">odd greedy expansion</a>?</li> <li>Do any <a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel numbers</a> exist?</li> <li>Do any odd <a href="/wiki/Noncototient" title="Noncototient">noncototients</a> exist?</li> <li>Do any odd <a href="/wiki/Weird_number" title="Weird number">weird numbers</a> exist?</li> <li>Do any <a href="/wiki/Superperfect_number#Generalizations" title="Superperfect number">(2, 5)-perfect numbers</a> exist?</li> <li>Do any <a href="/wiki/Generalized_taxicab_number" title="Generalized taxicab number">Taxicab(5, 2, n)</a> exist for n > 1?</li> <li>Is there a <a href="/wiki/Covering_system" title="Covering system">covering system</a> with odd distinct moduli?<a href="#cite_note-146">[146]</a></li> <li>Is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> a <a href="/wiki/Normal_number" title="Normal number">normal number</a> (i.e., is each digit 0–9 equally frequent)?<a href="#cite_note-147">[147]</a></li> <li>Are all <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic</a> numbers normal?</li> <li>Is 10 a <a href="/wiki/Solitary_number" class="mw-redirect" title="Solitary number">solitary number</a>?</li> <li>Can a 3×3 <a href="/wiki/Magic_square" title="Magic square">magic square</a> be constructed from 9 distinct perfect square numbers?<a href="#cite_note-148">[148]</a></li> <li>Find the value of the <a href="/wiki/De_Bruijn%E2%80%93Newman_constant" title="De Bruijn–Newman constant">De Bruijn–Newman constant</a>.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Additive_number_theory">Additive number theory</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Additive_number_theory" title="Additive number theory">Additive number theory</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Problems_involving_arithmetic_progressions" title="Problems involving arithmetic progressions">Problems involving arithmetic progressions</a></div> <ul><li><a href="/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions" title="Erdős conjecture on arithmetic progressions">Erdős conjecture on arithmetic progressions</a> that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progressions</a>.</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Tur%C3%A1n_conjecture_on_additive_bases" title="Erdős–Turán conjecture on additive bases">Erdős–Turán conjecture on additive bases</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is an <a href="/wiki/Additive_basis" title="Additive basis">additive basis</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">, then the number of ways that positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> can be expressed as the sum of two numbers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> must tend to infinity as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> tends to infinity.</li> <li><a href="/wiki/Gilbreath%27s_conjecture" title="Gilbreath's conjecture">Gilbreath's conjecture</a> on consecutive applications of the unsigned <a href="/wiki/Finite_difference" title="Finite difference">forward difference</a> operator to the sequence of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>.</li> <li><a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a>: every even natural number greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> is the sum of two <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>.</li> <li><a href="/wiki/Lander,_Parkin,_and_Selfridge_conjecture" title="Lander, Parkin, and Selfridge conjecture">Lander, Parkin, and Selfridge conjecture</a>: if the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-th powers of positive integers is equal to a different sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-th powers of positive integers, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+n\geq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>≥</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+n\geq k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742ac2b937fb0debcf8e32d6fb269935b2557740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.585ex; height:2.343ex;" alt="{\displaystyle m+n\geq k}">.</li> <li><a href="/wiki/Lemoine%27s_conjecture" title="Lemoine's conjecture">Lemoine's conjecture</a>: all odd integers greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> can be represented as the sum of an odd <a href="/wiki/Prime_number" title="Prime number">prime number</a> and an even <a href="/wiki/Semiprime" title="Semiprime">semiprime</a>.</li> <li><a href="/wiki/Minimum_overlap_problem" title="Minimum overlap problem">Minimum overlap problem</a> of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,2n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,2n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8e827c68dd8b99ef59ebf0b7d2fd930f9b70ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.223ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,2n\}}"></li> <li><a href="/wiki/Pollock%27s_conjectures" title="Pollock's conjectures">Pollock's conjectures</a></li> <li>Does every nonnegative integer appear in <a href="/wiki/Recam%C3%A1n%27s_sequence" title="Recamán's sequence">Recamán's sequence</a>?</li> <li><a href="/wiki/Skolem_problem" title="Skolem problem">Skolem problem</a>: can an algorithm determine if a <a href="/wiki/Constant-recursive_sequence" title="Constant-recursive sequence">constant-recursive sequence</a> contains a zero?</li> <li>The values of g(k) and G(k) in <a href="/wiki/Waring%27s_problem" title="Waring's problem">Waring's problem</a></li></ul> <ul><li>Do the <a href="/wiki/Ulam_number" title="Ulam number">Ulam numbers</a> have a positive density?</li></ul> <ul><li>Determine growth rate of rk(N) (see <a href="/wiki/Szemer%C3%A9di%27s_theorem" title="Szemerédi's theorem">Szemerédi's theorem</a>)</li></ul> <div class="mw-heading mw-heading4"><h4 id="Algebraic_number_theory">Algebraic number theory</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></div> <ul><li><a href="/wiki/Class_number_problem" title="Class number problem">Class number problem</a>: are there infinitely many <a href="/wiki/Class_number_problem#Real_quadratic_fields" title="Class number problem">real quadratic number fields</a> with <a href="/wiki/Unique_factorization" class="mw-redirect" title="Unique factorization">unique factorization</a>?</li> <li><a href="/wiki/Fontaine%E2%80%93Mazur_conjecture" title="Fontaine–Mazur conjecture">Fontaine–Mazur conjecture</a>: actually numerous conjectures, all proposed by <a href="/wiki/Jean-Marc_Fontaine" title="Jean-Marc Fontaine">Jean-Marc Fontaine</a> and <a href="/wiki/Barry_Mazur" title="Barry Mazur">Barry Mazur</a>.</li> <li><a href="/wiki/Gan%E2%80%93Gross%E2%80%93Prasad_conjecture" title="Gan–Gross–Prasad conjecture">Gan–Gross–Prasad conjecture</a>: a <a href="/wiki/Restricted_representation" title="Restricted representation">restriction</a> problem in <a href="/wiki/Representation_of_a_Lie_group" title="Representation of a Lie group">representation theory of real or p-adic Lie groups</a>.</li> <li><a href="/wiki/Greenberg%27s_conjectures" title="Greenberg's conjectures">Greenberg's conjectures</a></li> <li><a href="/wiki/Hermite%27s_problem" title="Hermite's problem">Hermite's problem</a>: is it possible, for any natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, to assign a sequence of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> to each <a href="/wiki/Real_number" title="Real number">real number</a> such that the sequence for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is eventually <a href="/wiki/Periodic_sequence" title="Periodic sequence">periodic</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">?</li> <li><a href="/wiki/Kummer%E2%80%93Vandiver_conjecture" title="Kummer–Vandiver conjecture">Kummer–Vandiver conjecture</a>: primes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> do not divide the <a href="/wiki/Ideal_class_group#Properties" title="Ideal class group">class number</a> of the maximal real <a href="/wiki/Field_extension" title="Field extension">subfield</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a>.</li> <li>Lang and Trotter's conjecture on <a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">supersingular primes</a> that the number of <a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">supersingular primes</a> less than a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is within a constant multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {X}}/\ln {X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>X</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {X}}/\ln {X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007945f34f075f0707882e5539ecab3669553a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.772ex; height:3.176ex;" alt="{\displaystyle {\sqrt {X}}/\ln {X}}"></li> <li><a href="/wiki/Selberg%27s_1/4_conjecture" title="Selberg's 1/4 conjecture">Selberg's 1/4 conjecture</a>: the <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> of the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> on <a href="/wiki/Maass_wave_form" title="Maass wave form">Maass wave forms</a> of <a href="/wiki/Congruence_subgroup" title="Congruence subgroup">congruence subgroups</a> are at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf1ef33695c3d98cb09f01e5700f927ce928c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/4}">.</li> <li><a href="/wiki/Stark_conjectures" title="Stark conjectures">Stark conjectures</a> (including <a href="/wiki/Brumer%E2%80%93Stark_conjecture" title="Brumer–Stark conjecture">Brumer–Stark conjecture</a>)</li></ul> <ul><li>Characterize all algebraic number fields that have some <a href="/wiki/Algebraic_number_field#Bases_for_number_fields" title="Algebraic number field">power basis</a>.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Computational_number_theory">Computational number theory</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></div> <ul><li>Can <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a> be done in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>?</li></ul> <div class="mw-heading mw-heading4"><h4 id="Diophantine_approximation_and_transcendental_number_theory">Diophantine approximation and transcendental number theory</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a> and <a href="/wiki/Transcendental_number_theory" title="Transcendental number theory">Transcendental number theory</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gamma-area.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Gamma-area.svg/220px-Gamma-area.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Gamma-area.svg/330px-Gamma-area.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Gamma-area.svg/440px-Gamma-area.svg.png 2x" data-file-width="720" data-file-height="480" /></a><figcaption>The area of the blue region converges to the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>, which may or may not be a rational number.</figcaption></figure> <ul><li><a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a> on the <a href="/wiki/Transcendence_degree" class="mw-redirect" title="Transcendence degree">transcendence degree</a> of certain <a href="/wiki/Field_extension" title="Field extension">field extensions</a> of the rational numbers.<a href="#cite_note-waldschmidt-149">[149]</a> In particular: Are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> <a href="/wiki/Algebraic_independence" title="Algebraic independence">algebraically independent</a>? Which nontrivial combinations of <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a> (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>+</mo> <mi>π</mi> <mo>,</mo> <mi>e</mi> <mi>π</mi> <mo>,</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf20dad9379caf3bf97a076e9d4fe37cefe20ebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.73ex; height:2.676ex;" alt="{\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{e}}">) are themselves transcendental?<a href="#cite_note-150">[150]</a><a href="#cite_note-151">[151]</a></li> <li>The <a href="/wiki/Four_exponentials_conjecture" title="Four exponentials conjecture">four exponentials conjecture</a>: the transcendence of at least one of four exponentials of combinations of irrationals<a href="#cite_note-waldschmidt-149">[149]</a></li> <li>Are <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler's constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> and <a href="/wiki/Catalan%27s_constant" title="Catalan's constant">Catalan's constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> irrational? Are they transcendental? Is <a href="/wiki/Ap%C3%A9ry%27s_constant" title="Apéry's constant">Apéry's constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3088978098c7b90b2754a9d9b0b994d873e1755c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.067ex; height:2.843ex;" alt="{\displaystyle \zeta (3)}"> transcendental?<a href="#cite_note-152">[152]</a><a href="#cite_note-:1-153">[153]</a></li> <li>Which transcendental numbers are <a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">(exponential) periods</a>?<a href="#cite_note-154">[154]</a></li> <li>How well can <a href="/wiki/Quadratic_equation" title="Quadratic equation">non-quadratic</a> irrational numbers be approximated? What is the <a href="/wiki/Irrationality_measure" title="Irrationality measure">irrationality measure</a> of specific (suspected) transcendental numbers such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }">?<a href="#cite_note-:1-153">[153]</a></li> <li>Which irrational numbers have <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fraction</a> terms whose <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> converges to <a href="/wiki/Khinchin%27s_constant" title="Khinchin's constant">Khinchin's constant</a>?<a href="#cite_note-155">[155]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Diophantine_equations">Diophantine equations</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a></div> <ul><li><a href="/wiki/Beal%27s_conjecture" class="mw-redirect" title="Beal's conjecture">Beal's conjecture</a>: for all integral solutions to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{x}+B^{y}=C^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{x}+B^{y}=C^{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e579ed4b1e64d713b6255681b9103dfc8bad8cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.467ex; height:2.509ex;" alt="{\displaystyle A^{x}+B^{y}=C^{z}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z>2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c693f1e3aa5b6f4053242e790610ecb31a93ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.902ex; height:2.509ex;" alt="{\displaystyle x,y,z>2}">, all three numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"> must share some prime factor.</li> <li><a href="/wiki/Congruent_number_problem" class="mw-redirect" title="Congruent number problem">Congruent number problem</a> (a corollary to <a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a>, per <a href="/wiki/Tunnell%27s_theorem" title="Tunnell's theorem">Tunnell's theorem</a>): determine precisely what rational numbers are <a href="/wiki/Congruent_number" title="Congruent number">congruent numbers</a>.</li> <li>Erdős–Moser problem: is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{1}+2^{1}=3^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{1}+2^{1}=3^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08bb8af942b066453ddbed89d7cab7e1665fcf31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.589ex; height:2.843ex;" alt="{\displaystyle 1^{1}+2^{1}=3^{1}}"> the only solution to the <a href="/wiki/Erd%C5%91s%E2%80%93Moser_equation" title="Erdős–Moser equation">Erdős–Moser equation</a>?</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Straus_conjecture" title="Erdős–Straus conjecture">Erdős–Straus conjecture</a>: for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}">, there are positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4/n=1/x+1/y+1/z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4/n=1/x+1/y+1/z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aabfae6433b32a07b54ae29bff11abae5d74830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.047ex; height:2.843ex;" alt="{\displaystyle 4/n=1/x+1/y+1/z}">.</li> <li><a href="/wiki/Fermat%E2%80%93Catalan_conjecture" title="Fermat–Catalan conjecture">Fermat–Catalan conjecture</a>: there are finitely many distinct solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{m},b^{n},c^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{m},b^{n},c^{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0d58c53a987d94141b810856ec91e0e18f5880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.094ex; height:3.176ex;" alt="{\displaystyle (a^{m},b^{n},c^{k})}"> to the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{m}+b^{n}=c^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{m}+b^{n}=c^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3123b1f3f56d4029e9308f91c9b00aa785ba985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.155ex; height:2.843ex;" alt="{\displaystyle a^{m}+b^{n}=c^{k}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> being positive <a href="/wiki/Coprime_integers" title="Coprime integers">coprime integers</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n,k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a958b5e16104a290f7982ca4425d60863f3e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.714ex; height:2.509ex;" alt="{\displaystyle m,n,k}"> being positive integers satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/m+1/n+1/k<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/m+1/n+1/k<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1406267891f5f8bbbd3c91777186ba90ef525bf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.563ex; height:2.843ex;" alt="{\displaystyle 1/m+1/n+1/k<1}">.</li> <li><a href="/wiki/Goormaghtigh_conjecture" title="Goormaghtigh conjecture">Goormaghtigh conjecture</a> on solutions to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769e849b912b88a8a073c4fa781d99fe761d3095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.541ex; height:2.843ex;" alt="{\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>y>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mi>y</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>y>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c75ca33e807a3dd892bc1e9157b119f58571d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.845ex; height:2.509ex;" alt="{\displaystyle x>y>1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n>2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1314eaa2adee834031142ea98977b9eeb4a6d2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.73ex; height:2.509ex;" alt="{\displaystyle m,n>2}">.</li> <li>The <a href="/wiki/Markov_number#Other_properties" title="Markov number">uniqueness conjecture for Markov numbers</a><a href="#cite_note-156">[156]</a> that every <a href="/wiki/Markov_number" title="Markov number">Markov number</a> is the largest number in exactly one normalized solution to the Markov <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a>.</li> <li><a href="/wiki/Pillai%27s_conjecture" class="mw-redirect" title="Pillai's conjecture">Pillai's conjecture</a>: for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}">, the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax^{m}-By^{n}=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mi>B</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{m}-By^{n}=C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24044dfffbbb6c9ab654c2730c437941bdc3b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.596ex; height:2.676ex;" alt="{\displaystyle Ax^{m}-By^{n}=C}"> has finitely many solutions when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6568e95b6bf8f39b7fd2c9b52b7b00ee124c6250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.009ex;" alt="{\displaystyle m,n}"> are not both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">.</li> <li>Which integers can be written as the <a href="/wiki/Sums_of_three_cubes" title="Sums of three cubes">sum of three perfect cubes</a>?<a href="#cite_note-157">[157]</a></li> <li><a href="/wiki/Sum_of_four_cubes_problem" title="Sum of four cubes problem">Can every integer be written as a sum of four perfect cubes?</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Prime_numbers">Prime numbers</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Prime_numbers" 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href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_conjectures" title="Template:Prime number conjectures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_conjectures" title="Template talk:Prime number conjectures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_conjectures" title="Special:EditPage/Template:Prime number conjectures"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_conjectures" style="font-size:114%;margin:0 4em">Prime number conjectures</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy%E2%80%93Littlewood_conjecture" class="mw-redirect" title="Hardy–Littlewood conjecture">Hardy–Littlewood</a> <ul><li><a href="/wiki/First_Hardy%E2%80%93Littlewood_conjecture" title="First Hardy–Littlewood conjecture">1st</a></li> <li><a href="/wiki/Second_Hardy%E2%80%93Littlewood_conjecture" title="Second Hardy–Littlewood conjecture">2nd</a></li></ul></li> <li><a href="/wiki/Agoh%E2%80%93Giuga_conjecture" title="Agoh–Giuga conjecture">Agoh–Giuga</a></li> <li><a href="/wiki/Andrica%27s_conjecture" title="Andrica's conjecture">Andrica's</a></li> <li><a href="/wiki/Artin%27s_conjecture_on_primitive_roots" title="Artin's conjecture on primitive roots">Artin's</a></li> <li><a href="/wiki/Bateman%E2%80%93Horn_conjecture" title="Bateman–Horn conjecture">Bateman–Horn</a></li> <li><a href="/wiki/Brocard%27s_conjecture" title="Brocard's conjecture">Brocard's</a></li> <li><a href="/wiki/Bunyakovsky_conjecture" title="Bunyakovsky conjecture">Bunyakovsky</a></li> <li><a href="/wiki/Chinese_hypothesis" title="Chinese hypothesis">Chinese hypothesis</a></li> <li><a href="/wiki/Cram%C3%A9r%27s_conjecture" title="Cramér's conjecture">Cramér's (Shanks')</a></li> <li><a href="/wiki/Dickson%27s_conjecture" title="Dickson's conjecture">Dickson's</a></li> <li><a href="/wiki/Elliott%E2%80%93Halberstam_conjecture" title="Elliott–Halberstam conjecture">Elliott–Halberstam</a></li> <li><a href="/wiki/Firoozbakht%27s_conjecture" title="Firoozbakht's conjecture">Firoozbakht's (Forgues', Nicholson's, Farhadian's)</a></li> <li><a href="/wiki/Gilbreath%27s_conjecture" title="Gilbreath's conjecture">Gilbreath's</a></li> <li><a href="/wiki/Grimm%27s_conjecture" title="Grimm's conjecture">Grimm's</a></li> <li><a href="/wiki/Landau%27s_problems" title="Landau's problems">Landau's problems</a> <ul><li><a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's</a> <ul><li><a href="/wiki/Goldbach%27s_weak_conjecture" title="Goldbach's weak conjecture">weak</a></li></ul></li> <li><a href="/wiki/Legendre%27s_conjecture" title="Legendre's conjecture">Legendre's</a></li> <li><a href="/wiki/Twin_prime_conjecture" class="mw-redirect" title="Twin prime conjecture">Twin prime</a></li></ul></li> <li><a href="/wiki/Legendre%27s_constant" title="Legendre's constant">Legendre's constant</a></li> <li><a href="/wiki/Lemoine%27s_conjecture" title="Lemoine's conjecture">Lemoine's</a></li> <li><a href="/wiki/Mersenne_conjectures" title="Mersenne conjectures">Mersenne</a></li> <li><a href="/wiki/Oppermann%27s_conjecture" title="Oppermann's conjecture">Oppermann's</a></li> <li><a href="/wiki/Polignac%27s_conjecture" title="Polignac's conjecture">Polignac's</a></li> <li><a href="/wiki/P%C3%B3lya_conjecture" title="Pólya conjecture">Pólya</a></li> <li><a href="/wiki/Schinzel%27s_hypothesis_H" title="Schinzel's hypothesis H">Schinzel's hypothesis H</a></li> <li><a href="/wiki/Waring%27s_prime_number_conjecture" title="Waring's prime number conjecture">Waring's prime number</a></li></ul> </div></td></tr></tbody></table></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Goldbach_partitions_of_the_even_integers_from_4_to_50_rev4b.svg" title="File:Goldbach partitions of the even integers from 4 to 50 rev4b.svg"><img resource="/wiki/File:Goldbach_partitions_of_the_even_integers_from_4_to_50_rev4b.svg" src="//upload.wikimedia.org/wikipedia/commons/d/dd/Goldbach_partitions_of_the_even_integers_from_4_to_28_300px.png" decoding="async" width="300" height="283" class="mw-file-element" data-file-width="300" data-file-height="283" /></a><figcaption><a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.</figcaption></figure> <ul><li><a href="/wiki/Agoh%E2%80%93Giuga_conjecture" title="Agoh–Giuga conjecture">Agoh–Giuga conjecture</a> on the <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is prime if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pB_{p-1}\equiv -1{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pB_{p-1}\equiv -1{\pmod {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45bbc5b944ef7488e3965ece35d3dddb7cc67654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:23.105ex; height:3.009ex;" alt="{\displaystyle pB_{p-1}\equiv -1{\pmod {p}}}"></li> <li><a href="/wiki/Agrawal%27s_conjecture" title="Agrawal's conjecture">Agrawal's conjecture</a> that given <a href="/wiki/Coprime_integers" title="Coprime integers">coprime positive integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>≡</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo>,</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96fe434f1db5b7fd76ff87c685469d9bf1f0d630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.413ex; height:2.843ex;" alt="{\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1}}}">, then either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is prime or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}\equiv 1{\pmod {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}\equiv 1{\pmod {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84397b34b97c6792e676740a268d1bd5efe0b37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.443ex; height:3.176ex;" alt="{\displaystyle n^{2}\equiv 1{\pmod {r}}}"></li> <li><a href="/wiki/Artin%27s_conjecture_on_primitive_roots" title="Artin's conjecture on primitive roots">Artin's conjecture on primitive roots</a> that if an integer is neither a perfect square nor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">, then it is a <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive root</a> modulo infinitely many <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></li> <li><a href="/wiki/Brocard%27s_conjecture" title="Brocard's conjecture">Brocard's conjecture</a>: there are always at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"> <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> between consecutive squares of prime numbers, aside from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd7711cd907a2d46557a410fb67fc0d84c52ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84cba38173d6b69364f2016245721c333282e0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 3^{2}}">.</li> <li><a href="/wiki/Bunyakovsky_conjecture" title="Bunyakovsky conjecture">Bunyakovsky conjecture</a>: if an integer-coefficient polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> has a positive leading coefficient, is irreducible over the integers, and has no common factors over all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is a positive integer, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is prime infinitely often.</li> <li><a href="/wiki/Catalan%27s_Mersenne_conjecture" class="mw-redirect" title="Catalan's Mersenne conjecture">Catalan's Mersenne conjecture</a>: some <a href="/wiki/Double_Mersenne_number#Catalan–Mersenne_number_conjecture" title="Double Mersenne number">Catalan–Mersenne number</a> is composite and thus all Catalan–Mersenne numbers are composite after some point.</li> <li><a href="/wiki/Dickson%27s_conjecture" title="Dickson's conjecture">Dickson's conjecture</a>: for a finite set of linear forms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>n</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb10dce43dd93cca89f82d0842c6f676434efb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.389ex; height:2.509ex;" alt="{\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n}"> with each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136d19bf28b138b6bc5a82f6145837f23e19253f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.058ex; height:2.509ex;" alt="{\displaystyle b_{i}\geq 1}">, there are infinitely many <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> for which all forms are <a href="/wiki/Prime_number" title="Prime number">prime</a>, unless there is some <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">congruence</a> condition preventing it.</li> <li>Dubner's conjecture: every even number greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4208}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4208</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4208}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf748b5a4d51034390cd771257f3996295c11d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.65ex; height:2.176ex;" alt="{\displaystyle 4208}"> is the sum of two <a href="/wiki/Prime_number" title="Prime number">primes</a> which both have a <a href="/wiki/Twin_prime" title="Twin prime">twin</a>.</li> <li><a href="/wiki/Elliott%E2%80%93Halberstam_conjecture" title="Elliott–Halberstam conjecture">Elliott–Halberstam conjecture</a> on the distribution of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> in <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progressions</a>.</li> <li><a href="/wiki/Powerful_number#Mathematical_properties" title="Powerful number">Erdős–Mollin–Walsh conjecture</a>: no three consecutive numbers are all <a href="/wiki/Powerful_number" title="Powerful number">powerful</a>.</li> <li><a href="/wiki/Feit%E2%80%93Thompson_conjecture" title="Feit–Thompson conjecture">Feit–Thompson conjecture</a>: for all distinct <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p^{q}-1)/(p-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p^{q}-1)/(p-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb2afa4999a4bef33e4de378e4a081b5b3b0830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.114ex; height:2.843ex;" alt="{\displaystyle (p^{q}-1)/(p-1)}"> does not divide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q^{p}-1)/(q-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q^{p}-1)/(q-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5a30551f59f2ed380015f1ea83cf5313b6e2174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.995ex; height:2.843ex;" alt="{\displaystyle (q^{p}-1)/(q-1)}"></li> <li>Fortune's conjecture that no <a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate number</a> is composite.</li> <li>The <a href="/wiki/Gaussian_moat" title="Gaussian moat">Gaussian moat</a> problem: is it possible to find an infinite sequence of distinct <a href="/wiki/Gaussian_prime_number" class="mw-redirect" title="Gaussian prime number">Gaussian prime numbers</a> such that the difference between consecutive numbers in the sequence is bounded?</li> <li><a href="/wiki/Gillies%27_conjecture" title="Gillies' conjecture">Gillies' conjecture</a> on the distribution of <a href="/wiki/Prime_number" title="Prime number">prime</a> divisors of <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne numbers</a>.</li> <li><a href="/wiki/Landau%27s_problems" title="Landau's problems">Landau's problems</a> <ul><li><a href="/wiki/Goldbach_conjecture" class="mw-redirect" title="Goldbach conjecture">Goldbach conjecture</a>: all even <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> are the sum of two <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>.</li> <li><a href="/wiki/Legendre%27s_conjecture" title="Legendre's conjecture">Legendre's conjecture</a>: for every positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, there is a prime between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660fbe855b0a0cbe51cad9de61f0fb05908c6661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.261ex; height:3.176ex;" alt="{\displaystyle (n+1)^{2}}">.</li> <li><a href="/wiki/Twin_prime#Twin_prime_conjecture" title="Twin prime">Twin prime conjecture</a>: there are infinitely many <a href="/wiki/Twin_prime" title="Twin prime">twin primes</a>.</li> <li>Are there infinitely many primes of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0744bc6c0cbdf56197698229110e29b06cb8a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.452ex; height:2.843ex;" alt="{\displaystyle n^{2}+1}">?</li></ul></li> <li>Problems associated to <a href="/wiki/Linnik%27s_theorem" title="Linnik's theorem">Linnik's theorem</a></li> <li><a href="/wiki/Mersenne_conjectures#New_Mersenne_conjecture" title="Mersenne conjectures">New Mersenne conjecture</a>: for any odd <a href="/wiki/Natural_number" title="Natural number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">, if any two of the three conditions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2^{k}\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2^{k}\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d108e28cc35bc9994dbcc21b2c53dd012b4b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.611ex; height:3.009ex;" alt="{\displaystyle p=2^{k}\pm 1}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=4^{k}\pm 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>±</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=4^{k}\pm 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b56dae2374f0118c3f64539d177c851db9c707c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.611ex; height:3.009ex;" alt="{\displaystyle p=4^{k}\pm 3}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{p}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c5977dbf385ba719fbb90f67b0a3d91e1da6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.224ex; height:2.509ex;" alt="{\displaystyle 2^{p}-1}"> is prime, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{p}+1)/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2^{p}+1)/3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e026ad87a946d1d5ccefde798bae9ca7ca6bd3b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.359ex; height:2.843ex;" alt="{\displaystyle (2^{p}+1)/3}"> is prime are true, then the third condition is also true.</li> <li><a href="/wiki/Polignac%27s_conjecture" title="Polignac's conjecture">Polignac's conjecture</a>: for all positive even numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, there are infinitely many <a href="/wiki/Prime_gap" title="Prime gap">prime gaps</a> of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.</li> <li><a href="/wiki/Schinzel%27s_hypothesis_H" title="Schinzel's hypothesis H">Schinzel's hypothesis H</a> that for every finite collection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{1},\ldots ,f_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f_{1},\ldots ,f_{k}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17b7f693d1809e863e4b7c1ee4cc3cfc42f75fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.925ex; height:2.843ex;" alt="{\displaystyle \{f_{1},\ldots ,f_{k}\}}"> of nonconstant <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomials</a> over the integers with positive leading coefficients, either there are infinitely many positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(n),\ldots ,f_{k}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(n),\ldots ,f_{k}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d3f34c43079ec67cf5e477020f3f891f9dbef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.008ex; height:2.843ex;" alt="{\displaystyle f_{1}(n),\ldots ,f_{k}(n)}"> are all <a href="/wiki/Prime_number" title="Prime number">primes</a>, or there is some fixed divisor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f27527902d05e4c32bcbe28d425d7790f8ae191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>1}"> which, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, divides some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58335d44ef6dbb4ce8678ce00f2bdfb2daee93aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.143ex; height:2.843ex;" alt="{\displaystyle f_{i}(n)}">.</li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Selfridge's conjecture</a>: is 78,557 the lowest <a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński number</a>?</li> <li>Does the <a href="/wiki/Wolstenholme%27s_theorem#The_converse_as_a_conjecture" title="Wolstenholme's theorem">converse of Wolstenholme's theorem</a> hold for all natural numbers?</li></ul> <ul><li>Are all <a href="/wiki/Euclid_number" title="Euclid number">Euclid numbers</a> <a href="/wiki/Square-free_integer" title="Square-free integer">square-free</a>?</li> <li>Are all <a href="/wiki/Fermat_number" title="Fermat number">Fermat numbers</a> <a href="/wiki/Square-free_integer" title="Square-free integer">square-free</a>?</li> <li>Are all <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne numbers</a> of prime index <a href="/wiki/Square-free_integer" title="Square-free integer">square-free</a>?</li> <li>Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?</li> <li>Are there any <a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun primes</a>?</li> <li>Are there any <a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich primes</a> in base 47?</li> <li>Are there infinitely many <a href="/wiki/Balanced_prime" title="Balanced prime">balanced primes</a>?</li> <li>Are there infinitely many Carol primes?</li> <li>Are there infinitely many <a href="/wiki/Cluster_prime" title="Cluster prime">cluster primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Cousin_prime" title="Cousin prime">cousin primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Cullen_number" title="Cullen number">Cullen primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Euclid_number" title="Euclid number">Euclid primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Euclid_number#Generalization" title="Euclid number">Kummer primes</a>?</li> <li>Are there infinitely many Kynea primes?</li> <li>Are there infinitely many <a href="/wiki/Lucas_number#Lucas_primes" title="Lucas number">Lucas primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne primes</a> (<a href="/wiki/Lenstra%E2%80%93Pomerance%E2%80%93Wagstaff_conjecture" class="mw-redirect" title="Lenstra–Pomerance–Wagstaff conjecture">Lenstra–Pomerance–Wagstaff conjecture</a>); equivalently, infinitely many even <a href="/wiki/Perfect_number" title="Perfect number">perfect numbers</a>?</li> <li>Are there infinitely many <a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Palindromic_prime" title="Palindromic prime">palindromic primes</a> to every base?</li> <li>Are there infinitely many <a href="/wiki/Pell_number" title="Pell number">Pell primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Prime_quadruplet" title="Prime quadruplet">prime quadruplets</a>?</li> <li>Are there infinitely many <a href="/wiki/Prime_triplet" title="Prime triplet">prime triplets</a>?</li> <li>Are there infinitely many <a href="/wiki/Regular_prime" title="Regular prime">regular primes</a>, and if so is their relative density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a56ec5c5b26f80a643b22ec4ea800cafc69ae14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.06ex; height:2.843ex;" alt="{\displaystyle e^{-1/2}}">?</li> <li>Are there infinitely many <a href="/wiki/Sexy_prime" title="Sexy prime">sexy primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Safe_and_Sophie_Germain_primes" title="Safe and Sophie Germain primes">safe and Sophie Germain primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Wilson_prime" title="Wilson prime">Wilson primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme primes</a>?</li> <li>Are there infinitely many <a href="/wiki/Woodall_number#Woodall_primes" title="Woodall number">Woodall primes</a>?</li> <li>Can a prime p satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0c2e45a693d9b8fdac3c5947e38fb67e4e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.491ex; height:3.176ex;" alt="{\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{p-1}\equiv 1{\pmod {p^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{p-1}\equiv 1{\pmod {p^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7897724adaf6a14eca67dd4855d46cdc89b278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.491ex; height:3.176ex;" alt="{\displaystyle 3^{p-1}\equiv 1{\pmod {p^{2}}}}"> simultaneously?<a href="#cite_note-158">[158]</a></li> <li>Does every prime number appear in the <a href="/wiki/Euclid%E2%80%93Mullin_sequence" title="Euclid–Mullin sequence">Euclid–Mullin sequence</a>?</li> <li>What is the smallest <a href="/wiki/Skewes%27s_number" title="Skewes's number">Skewes's number</a>?</li> <li>For any given integer a > 0, are there infinitely many <a href="/wiki/Lucas%E2%80%93Wieferich_prime" class="mw-redirect" title="Lucas–Wieferich prime">Lucas–Wieferich primes</a> associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)</li> <li>For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?<a href="#cite_note-159">[159]</a></li> <li>For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?</li> <li>For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many <a href="/wiki/Repunit" title="Repunit">repunit</a> primes to base b?</li> <li>For any given integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1,b\geq 2,c\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> <mi>c</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1,b\geq 2,c\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef9f484eb8ace7fc5baf3c8be4bb8c8db05999c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.066ex; height:2.676ex;" alt="{\displaystyle k\geq 1,b\geq 2,c\neq 0}">, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k\times b^{n}+c)/{\text{gcd}}(k+c,b-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>×</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>gcd</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k\times b^{n}+c)/{\text{gcd}}(k+c,b-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0b9d7e3d49af673d4c4ecd8b5bae4146ceb0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.476ex; height:2.843ex;" alt="{\displaystyle (k\times b^{n}+c)/{\text{gcd}}(k+c,b-1)}"> with integer n ≥ 1?</li> <li>Is every <a href="/wiki/Fermat_number" title="Fermat number">Fermat number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27f57a4191be088259902a790ef2fb093ffb812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.184ex; height:2.843ex;" alt="{\displaystyle 2^{2^{n}}+1}"> composite for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b13dc8b113121cdaf76a723a61aa4f8be1468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>4}">?</li> <li>Is 509,203 the lowest <a href="/wiki/Riesel_number" title="Riesel number">Riesel number</a>?</li></ul> <div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Set_theory" title="Set theory">Set theory</a></div> Note: These conjectures are about <a href="/wiki/Model_theory" title="Model theory">models</a> of <a href="/wiki/Zermelo-Frankel_set_theory" class="mw-redirect" title="Zermelo-Frankel set theory">Zermelo-Frankel set theory</a> with <a href="/wiki/Axiom_of_choice" title="Axiom of choice">choice</a>, and may not be able to be expressed in models of other set theories such as the various <a href="/wiki/Constructive_set_theory" title="Constructive set theory">constructive set theories</a> or <a href="/wiki/Non-wellfounded_set_theory" class="mw-redirect" title="Non-wellfounded set theory">non-wellfounded set theory</a>. <ul><li>(<a href="/wiki/W._Hugh_Woodin" title="W. Hugh Woodin">Woodin</a>) Does the <a href="/wiki/Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a> below a <a href="/wiki/Strongly_compact_cardinal" title="Strongly compact cardinal">strongly compact cardinal</a> imply the <a href="/wiki/Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a> everywhere?</li> <li>Does the <a href="/wiki/Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a> entail <a href="/wiki/Diamondsuit" class="mw-redirect" title="Diamondsuit"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">♢</mi> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>cf</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </msubsup> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/059f357bf72a29a78ecf0dfa53eace8216f71203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.036ex; height:4.009ex;" alt="{\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})}"></a> for every <a href="/wiki/Singular_cardinal" class="mw-redirect" title="Singular cardinal">singular cardinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">?</li> <li>Does the <a href="/wiki/Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a> imply the existence of an <a href="/wiki/Suslin_tree" title="Suslin tree">ℵ2-Suslin tree</a>?</li> <li>If ℵω is a strong limit cardinal, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mrow> </msup> <mo><</mo> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d31494986f0ad425a2433e2876c2cedb139576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.998ex; height:3.343ex;" alt="{\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{1}}}"> (see <a href="/wiki/Singular_cardinals_hypothesis" title="Singular cardinals hypothesis">Singular cardinals hypothesis</a>)? The best bound, ℵω4, was obtained by <a href="/wiki/Saharon_Shelah" title="Saharon Shelah">Shelah</a> using his <a href="/wiki/PCF_theory" title="PCF theory">PCF theory</a>.</li> <li>The problem of finding the ultimate <a href="/wiki/Core_model" title="Core model">core model</a>, one that contains all <a href="/wiki/Large_cardinal_property" class="mw-redirect" title="Large cardinal property">large cardinals</a>.</li> <li><a href="/wiki/W._Hugh_Woodin" title="W. Hugh Woodin">Woodin's</a> <a href="/wiki/%CE%A9-logic" title="Ω-logic">Ω-conjecture</a>: if there is a <a href="/wiki/Class_(set_theory)" title="Class (set theory)">proper class</a> of <a href="/wiki/Woodin_cardinal" title="Woodin cardinal">Woodin cardinals</a>, then <a href="/wiki/%CE%A9-logic" title="Ω-logic">Ω-logic</a> satisfies an analogue of <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a>.</li> <li>Does the <a href="/wiki/Consistency" title="Consistency">consistency</a> of the existence of a <a href="/wiki/Strongly_compact_cardinal" title="Strongly compact cardinal">strongly compact cardinal</a> imply the consistent existence of a <a href="/wiki/Supercompact_cardinal" title="Supercompact cardinal">supercompact cardinal</a>?</li> <li>Does there exist a <a href="/wiki/J%C3%B3nsson_cardinal" title="Jónsson cardinal">Jónsson algebra</a> on ℵω?</li> <li>Is OCA (the <a href="/wiki/Open_coloring_axiom" title="Open coloring axiom">open coloring axiom</a>) consistent with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}>\aleph _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mo>></mo> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}>\aleph _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d33ea7f548a17f2b49ecbb8bd45b6954a5da37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.804ex; height:3.009ex;" alt="{\displaystyle 2^{\aleph _{0}}>\aleph _{2}}">?</li> <li><a href="/wiki/Reinhardt_cardinal" title="Reinhardt cardinal">Reinhardt cardinals</a>: Without assuming the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, can a <a href="/wiki/Reinhardt_cardinal" title="Reinhardt cardinal">nontrivial elementary embedding</a> V→V exist?</li></ul> <div class="mw-heading mw-heading3"><h3 id="Topology">Topology</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Topology" title="Topology">Topology</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ochiai_unknot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Ochiai_unknot.svg/220px-Ochiai_unknot.svg.png" decoding="async" width="220" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Ochiai_unknot.svg/330px-Ochiai_unknot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Ochiai_unknot.svg/440px-Ochiai_unknot.svg.png 2x" data-file-width="868" data-file-height="682" /></a><figcaption>The <a href="/wiki/Unknotting_problem" title="Unknotting problem">unknotting problem</a> asks whether there is an efficient algorithm to identify when the shape presented in a <a href="/wiki/Knot_diagram" class="mw-redirect" title="Knot diagram">knot diagram</a> is actually the <a href="/wiki/Unknot" title="Unknot">unknot</a>.</figcaption></figure> <ul><li><a href="/wiki/Baum%E2%80%93Connes_conjecture" title="Baum–Connes conjecture">Baum–Connes conjecture</a>: the <a href="/wiki/Baum%E2%80%93Connes_conjecture#Formulation" title="Baum–Connes conjecture">assembly map</a> is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>.</li> <li><a href="/wiki/Berge_knot" title="Berge knot">Berge conjecture</a> that the only <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a> in the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> which admit <a href="/wiki/Lens_space" title="Lens space">lens space</a> <a href="/wiki/Dehn_surgery" title="Dehn surgery">surgeries</a> are <a href="/wiki/Berge_knot" title="Berge knot">Berge knots</a>.</li> <li><a href="/wiki/Bing%E2%80%93Borsuk_conjecture" title="Bing–Borsuk conjecture">Bing–Borsuk conjecture</a>: every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous</a> <a href="/wiki/Retraction_(topology)" title="Retraction (topology)">absolute neighborhood retract</a> is a <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a>.</li> <li><a href="/wiki/Borel_conjecture" title="Borel conjecture">Borel conjecture</a>: <a href="/wiki/Aspherical_space" title="Aspherical space">aspherical</a> <a href="/wiki/Closed_manifold" title="Closed manifold">closed manifolds</a> are determined up to <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> by their <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental groups</a>.</li> <li><a href="/wiki/Halperin_conjecture" title="Halperin conjecture">Halperin conjecture</a> on rational <a href="/wiki/Serre_spectral_sequence" title="Serre spectral sequence">Serre spectral sequences</a> of certain <a href="/wiki/Fibration" title="Fibration">fibrations</a>.</li> <li><a href="/wiki/Hilbert%E2%80%93Smith_conjecture" title="Hilbert–Smith conjecture">Hilbert–Smith conjecture</a>: if a <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact</a> <a href="/wiki/Topological_group" title="Topological group">topological group</a> has a <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, <a href="/wiki/Group_action#Remarkable_properties_of_actions" title="Group action">faithful group action</a> on a <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifold</a>, then the group must be a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>.</li> <li>Mazur's conjectures<a href="#cite_note-160">[160]</a></li> <li><a href="/wiki/Novikov_conjecture" title="Novikov conjecture">Novikov conjecture</a> on the <a href="/wiki/Homotopy#Invariance" title="Homotopy">homotopy invariance</a> of certain <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> in the <a href="/wiki/Pontryagin_class" title="Pontryagin class">Pontryagin classes</a> of a <a href="/wiki/Manifold" title="Manifold">manifold</a>, arising from the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a>.</li> <li><a href="/wiki/Quadrisecant" title="Quadrisecant">Quadrisecants</a> of <a href="/wiki/Wild_knot" title="Wild knot">wild knots</a>: it has been conjectured that wild knots always have infinitely many quadrisecants.<a href="#cite_note-161">[161]</a></li> <li><a href="/wiki/Ravenel_conjectures" class="mw-redirect" title="Ravenel conjectures">Telescope conjecture</a>: the last of <a href="/wiki/Ravenel%27s_conjectures" title="Ravenel's conjectures">Ravenel's conjectures</a> in <a href="/wiki/Stable_homotopy_theory" title="Stable homotopy theory">stable homotopy theory</a> to be resolved.<a href="#cite_note-163">[a]</a></li> <li><a href="/wiki/Unknotting_problem" title="Unknotting problem">Unknotting problem</a>: can <a href="/wiki/Unknot" title="Unknot">unknots</a> be recognized in <a href="/wiki/Time_complexity#Polynomial_time" title="Time complexity">polynomial time</a>?</li> <li><a href="/wiki/Volume_conjecture" title="Volume conjecture">Volume conjecture</a> relating <a href="/wiki/Quantum_invariant" title="Quantum invariant">quantum invariants</a> of <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a> to the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a> of their <a href="/wiki/Knot_complement" title="Knot complement">knot complements</a>.</li> <li><a href="/wiki/Whitehead_conjecture" title="Whitehead conjecture">Whitehead conjecture</a>: every <a href="/wiki/Connectedness" title="Connectedness">connected</a> <a href="/wiki/CW_complex#Inductive_construction_of_CW_complexes" title="CW complex">subcomplex</a> of a two-dimensional <a href="/wiki/Aspherical_space" title="Aspherical space">aspherical</a> <a href="/wiki/CW_complex" title="CW complex">CW complex</a> is aspherical.</li> <li><a href="/wiki/Zeeman_conjecture" title="Zeeman conjecture">Zeeman conjecture</a>: given a finite <a href="/wiki/Contractible_space" title="Contractible space">contractible</a> two-dimensional <a href="/wiki/CW_complex" title="CW complex">CW complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, is the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\times [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\times [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e2fb8d4b78388febd562c9e819d1d82dedd778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.559ex; height:2.843ex;" alt="{\displaystyle K\times [0,1]}"> <a href="/wiki/Collapse_(topology)" title="Collapse (topology)">collapsible</a>?</li></ul> <div class="mw-heading mw-heading2"><h2 id="Problems_solved_since_1995">Problems solved since 1995</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ricci_flow.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Ricci_flow.png/220px-Ricci_flow.png" decoding="async" width="220" height="405" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Ricci_flow.png/330px-Ricci_flow.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/52/Ricci_flow.png 2x" data-file-width="350" data-file-height="644" /></a><figcaption><a href="/wiki/Ricci_flow" title="Ricci flow">Ricci flow</a>, here illustrated with a 2D manifold, was the key tool in <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>'s <a href="/wiki/Poincar%C3%A9_conjecture#Solution" title="Poincaré conjecture">solution of the Poincaré conjecture</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Algebra_2">Algebra</h3></div> <ul><li><a href="/wiki/Uniform_boundedness_conjecture_for_rational_points#Mazur's_conjecture_B" title="Uniform boundedness conjecture for rational points">Mazur's conjecture B</a> (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020)<a href="#cite_note-164">[163]</a></li> <li><a href="/wiki/Suita_conjecture" title="Suita conjecture">Suita conjecture</a> (Qi'an Guan and <a href="/wiki/Xiangyu_Zhou" title="Xiangyu Zhou">Xiangyu Zhou</a>, 2015) <a href="#cite_note-165">[164]</a></li> <li><a href="/wiki/Torsion_conjecture" title="Torsion conjecture">Torsion conjecture</a> (<a href="/wiki/Lo%C3%AFc_Merel" title="Loïc Merel">Loïc Merel</a>, 1996)<a href="#cite_note-166">[165]</a></li> <li><a href="/wiki/Carlitz%E2%80%93Wan_conjecture" title="Carlitz–Wan conjecture">Carlitz–Wan conjecture</a> (<a href="/wiki/Hendrik_Lenstra" title="Hendrik Lenstra">Hendrik Lenstra</a>, 1995)<a href="#cite_note-167">[166]</a></li> <li><a href="/wiki/Serre%27s_multiplicity_conjectures#Nonnegativity" title="Serre's multiplicity conjectures">Serre's nonnegativity conjecture</a> (<a href="/wiki/Ofer_Gabber" title="Ofer Gabber">Ofer Gabber</a>, 1995)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Analysis_2">Analysis</h3></div> <ul><li><a href="/wiki/Kadison%E2%80%93Singer_problem" title="Kadison–Singer problem">Kadison–Singer problem</a> (<a href="/wiki/Adam_Marcus_(mathematician)" title="Adam Marcus (mathematician)">Adam Marcus</a>, <a href="/wiki/Daniel_Spielman" title="Daniel Spielman">Daniel Spielman</a> and <a href="/wiki/Nikhil_Srivastava" title="Nikhil Srivastava">Nikhil Srivastava</a>, 2013)<a href="#cite_note-Casazza2006-168">[167]</a><a href="#cite_note-SIAM02.2014-169">[168]</a> (and the <a href="/wiki/Hans_Georg_Feichtinger#Feichtinger's_conjecture" title="Hans Georg Feichtinger">Feichtinger's conjecture</a>, Anderson's paving conjectures, Weaver's discrepancy theoretic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KS_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KS_{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb20dd98814b07bb2a5d4518b99080553dbd7132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.465ex; height:2.509ex;" alt="{\displaystyle KS_{r}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KS'_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KS'_{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8930fb054a722a575e3ab75ca904c6a8240838d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.465ex; height:2.509ex;" alt="{\displaystyle KS'_{r}}"> conjectures, Bourgain-Tzafriri conjecture and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\epsilon }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d43f763cabc17610e01044eee5aa5f356af298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.664ex; height:2.509ex;" alt="{\displaystyle R_{\epsilon }}">-conjecture)</li> <li><a href="/wiki/Ahlfors_measure_conjecture" title="Ahlfors measure conjecture">Ahlfors measure conjecture</a> (<a href="/wiki/Ian_Agol" title="Ian Agol">Ian Agol</a>, 2004)<a href="#cite_note-Agol-170">[169]</a></li> <li><a href="/wiki/Gradient_conjecture" title="Gradient conjecture">Gradient conjecture</a> (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)<a href="#cite_note-171">[170]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Combinatorics_2">Combinatorics</h3></div> <ul><li><a href="/wiki/Erd%C5%91s_sumset_conjecture" title="Erdős sumset conjecture">Erdős sumset conjecture</a> (Joel Moreira, Florian Richter, Donald Robertson, 2018)<a href="#cite_note-172">[171]</a></li> <li><a href="/wiki/Simplicial_sphere" title="Simplicial sphere">McMullen's g-conjecture</a> on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)<a href="#cite_note-173">[172]</a><a href="#cite_note-174">[173]</a></li> <li><a href="/wiki/Hirsch_conjecture" title="Hirsch conjecture">Hirsch conjecture</a> (<a href="/wiki/Francisco_Santos_Leal" title="Francisco Santos Leal">Francisco Santos Leal</a>, 2010)<a href="#cite_note-175">[174]</a><a href="#cite_note-176">[175]</a></li> <li><a href="/wiki/Ira_Gessel#Gessel's_lattice_path_conjecture" title="Ira Gessel">Gessel's lattice path conjecture</a> (<a href="/wiki/Manuel_Kauers" title="Manuel Kauers">Manuel Kauers</a>, <a href="/wiki/Christoph_Koutschan" title="Christoph Koutschan">Christoph Koutschan</a>, and <a href="/wiki/Doron_Zeilberger" title="Doron Zeilberger">Doron Zeilberger</a>, 2009)<a href="#cite_note-177">[176]</a></li> <li><a href="/wiki/Stanley%E2%80%93Wilf_conjecture" title="Stanley–Wilf conjecture">Stanley–Wilf conjecture</a> (<a href="/wiki/G%C3%A1bor_Tardos" title="Gábor Tardos">Gábor Tardos</a> and <a href="/wiki/Adam_Marcus_(mathematician)" title="Adam Marcus (mathematician)">Adam Marcus</a>, 2004)<a href="#cite_note-178">[177]</a> (and also the Alon–Friedgut conjecture)</li> <li><a href="/wiki/Kemnitz%27s_conjecture" title="Kemnitz's conjecture">Kemnitz's conjecture</a> (<a href="/wiki/Christian_Reiher" title="Christian Reiher">Christian Reiher</a>, 2003, Carlos di Fiore, 2003)<a href="#cite_note-179">[178]</a></li> <li><a href="/wiki/Cameron%E2%80%93Erd%C5%91s_conjecture" title="Cameron–Erdős conjecture">Cameron–Erdős conjecture</a> (<a href="/wiki/Ben_J._Green" class="mw-redirect" title="Ben J. Green">Ben J. Green</a>, 2003, Alexander Sapozhenko, 2003)<a href="#cite_note-180">[179]</a><a href="#cite_note-181">[180]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Dynamical_systems_2">Dynamical systems</h3></div> <ul><li><a href="/wiki/Zimmer%27s_conjecture" title="Zimmer's conjecture">Zimmer's conjecture</a> (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017)<a href="#cite_note-182">[181]</a></li> <li><a href="/wiki/Painlev%C3%A9_conjecture" title="Painlevé conjecture">Painlevé conjecture</a> (Jinxin Xue, 2014)<a href="#cite_note-Xue1-183">[182]</a><a href="#cite_note-Xue2-184">[183]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Game_theory">Game theory</h3></div> <ul><li>Existence of a non-terminating game of <a href="/wiki/Beggar-my-neighbour" title="Beggar-my-neighbour">beggar-my-neighbour</a> (Brayden Casella, 2024)<a href="#cite_note-185">[184]</a></li> <li>The <a href="/wiki/Angel_problem" title="Angel problem">angel problem</a> (Various independent proofs, 2006)<a href="#cite_note-186">[185]</a><a href="#cite_note-187">[186]</a><a href="#cite_note-188">[187]</a><a href="#cite_note-189">[188]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Geometry_2">Geometry</h3></div> <div class="mw-heading mw-heading4"><h4 id="21st_century">21st century</h4></div> <ul><li><a href="/wiki/Einstein_problem" title="Einstein problem">Einstein problem</a> (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024)<a href="#cite_note-190">[189]</a></li> <li><a href="/w/index.php?title=Maximal_rank_conjecture&action=edit&redlink=1" class="new" title="Maximal rank conjecture (page does not exist)">Maximal rank conjecture</a> (Eric Larson, 2018)<a href="#cite_note-191">[190]</a></li> <li><a href="/wiki/Weibel%27s_conjecture" title="Weibel's conjecture">Weibel's conjecture</a> (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)<a href="#cite_note-192">[191]</a></li> <li><a href="/wiki/Yau%27s_conjecture" title="Yau's conjecture">Yau's conjecture</a> (<a href="/wiki/Antoine_Song" title="Antoine Song">Antoine Song</a>, 2018)<a href="#cite_note-193">[192]</a><a href="#cite_note-194">[193]</a></li> <li><a href="/wiki/Pentagonal_tiling" title="Pentagonal tiling">Pentagonal tiling</a> (Michaël Rao, 2017)<a href="#cite_note-195">[194]</a></li> <li><a href="/wiki/Willmore_conjecture" title="Willmore conjecture">Willmore conjecture</a> (<a href="/wiki/Fernando_Cod%C3%A1_Marques" title="Fernando Codá Marques">Fernando Codá Marques</a> and <a href="/wiki/Andr%C3%A9_Neves" title="André Neves">André Neves</a>, 2012)<a href="#cite_note-196">[195]</a></li> <li><a href="/wiki/Erd%C5%91s_distinct_distances_problem" title="Erdős distinct distances problem">Erdős distinct distances problem</a> (<a href="/wiki/Larry_Guth" title="Larry Guth">Larry Guth</a>, <a href="/wiki/Nets_Katz" title="Nets Katz">Nets Hawk Katz</a>, 2011)<a href="#cite_note-197">[196]</a></li> <li><a href="/wiki/Squaring_the_plane" class="mw-redirect" title="Squaring the plane">Heterogeneous tiling conjecture (squaring the plane)</a> (Frederick V. Henle and James M. Henle, 2008)<a href="#cite_note-198">[197]</a></li> <li><a href="/wiki/Tameness_conjecture" class="mw-redirect" title="Tameness conjecture">Tameness conjecture</a> (<a href="/wiki/Ian_Agol" title="Ian Agol">Ian Agol</a>, 2004)<a href="#cite_note-Agol-170">[169]</a></li> <li><a href="/wiki/Ending_lamination_theorem" title="Ending lamination theorem">Ending lamination theorem</a> (<a href="/wiki/Jeffrey_Brock" title="Jeffrey Brock">Jeffrey F. Brock</a>, <a href="/wiki/Richard_Canary" title="Richard Canary">Richard D. Canary</a>, <a href="/wiki/Yair_Minsky" title="Yair Minsky">Yair N. Minsky</a>, 2004)<a href="#cite_note-199">[198]</a></li> <li><a href="/wiki/Carpenter%27s_rule_problem" title="Carpenter's rule problem">Carpenter's rule problem</a> (<a href="/wiki/Robert_Connelly" title="Robert Connelly">Robert Connelly</a>, <a href="/wiki/Erik_Demaine" title="Erik Demaine">Erik Demaine</a>, Günter Rote, 2003)<a href="#cite_note-200">[199]</a></li> <li><a href="/wiki/Lambda_g_conjecture" title="Lambda g conjecture">Lambda g conjecture</a> (Carel Faber and <a href="/wiki/Rahul_Pandharipande" title="Rahul Pandharipande">Rahul Pandharipande</a>, 2003)<a href="#cite_note-201">[200]</a></li> <li><a href="/wiki/Nagata%27s_conjecture" title="Nagata's conjecture">Nagata's conjecture</a> (Ivan Shestakov, Ualbai Umirbaev, 2003)<a href="#cite_note-202">[201]</a></li> <li><a href="/wiki/Double_bubble_conjecture" class="mw-redirect" title="Double bubble conjecture">Double bubble conjecture</a> (<a href="/wiki/Michael_Hutchings_(mathematician)" title="Michael Hutchings (mathematician)">Michael Hutchings</a>, <a href="/wiki/Frank_Morgan_(mathematician)" title="Frank Morgan (mathematician)">Frank Morgan</a>, Manuel Ritoré, Antonio Ros, 2002)<a href="#cite_note-203">[202]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="20th_century">20th century</h4></div> <ul><li><a href="/wiki/Honeycomb_conjecture" title="Honeycomb conjecture">Honeycomb conjecture</a> (<a href="/wiki/Thomas_Callister_Hales" title="Thomas Callister Hales">Thomas Callister Hales</a>, 1999)<a href="#cite_note-204">[203]</a></li> <li><a href="/wiki/Lange%27s_conjecture" title="Lange's conjecture">Lange's conjecture</a> (<a href="/wiki/Montserrat_Teixidor_i_Bigas" title="Montserrat Teixidor i Bigas">Montserrat Teixidor i Bigas</a> and Barbara Russo, 1999)<a href="#cite_note-205">[204]</a></li> <li><a href="/wiki/Bogomolov_conjecture" title="Bogomolov conjecture">Bogomolov conjecture</a> (<a href="/wiki/Emmanuel_Ullmo" title="Emmanuel Ullmo">Emmanuel Ullmo</a>, 1998, <a href="/wiki/Shou-Wu_Zhang" title="Shou-Wu Zhang">Shou-Wu Zhang</a>, 1998)<a href="#cite_note-206">[205]</a><a href="#cite_note-207">[206]</a></li> <li><a href="/wiki/Kepler_conjecture" title="Kepler conjecture">Kepler conjecture</a> (Samuel Ferguson, <a href="/wiki/Thomas_Callister_Hales" title="Thomas Callister Hales">Thomas Callister Hales</a>, 1998)<a href="#cite_note-208">[207]</a></li> <li><a href="/wiki/Dodecahedral_conjecture" title="Dodecahedral conjecture">Dodecahedral conjecture</a> (<a href="/wiki/Thomas_Callister_Hales" title="Thomas Callister Hales">Thomas Callister Hales</a>, Sean McLaughlin, 1998)<a href="#cite_note-209">[208]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graph_theory_2">Graph theory</h3></div> <ul><li><a href="/wiki/Kahn%E2%80%93Kalai_conjecture" title="Kahn–Kalai conjecture">Kahn–Kalai conjecture</a> (<a href="/wiki/Jinyoung_Park_(mathematician)" title="Jinyoung Park (mathematician)">Jinyoung Park</a> and Huy Tuan Pham, 2022)<a href="#cite_note-210">[209]</a></li> <li><a href="/wiki/Blankenship%E2%80%93Oporowski_conjecture" class="mw-redirect" title="Blankenship–Oporowski conjecture">Blankenship–Oporowski conjecture</a> on the book thickness of subdivisions (<a href="/wiki/Vida_Dujmovi%C4%87" title="Vida Dujmović">Vida Dujmović</a>, <a href="/wiki/David_Eppstein" title="David Eppstein">David Eppstein</a>, Robert Hickingbotham, <a href="/wiki/Pat_Morin" title="Pat Morin">Pat Morin</a>, and <a href="/wiki/David_Wood_(mathematician)" title="David Wood (mathematician)">David Wood</a>, 2021)<a href="#cite_note-211">[210]</a></li> <li><a href="/wiki/Graceful_labeling" title="Graceful labeling">Ringel's conjecture</a> that the complete graph <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc21967f557f1281e605947db7b1d0018012388b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.114ex; height:2.509ex;" alt="{\displaystyle K_{2n+1}}"> can be decomposed into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca410f731fe4c7c444330343afb1d1850eadaea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n+1}"> copies of any tree with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> edges (Richard Montgomery, <a href="/wiki/Benny_Sudakov" title="Benny Sudakov">Benny Sudakov</a>, Alexey Pokrovskiy, 2020)<a href="#cite_note-212">[211]</a><a href="#cite_note-213">[212]</a></li> <li>Disproof of <a href="/wiki/Hedetniemi%27s_conjecture" title="Hedetniemi's conjecture">Hedetniemi's conjecture</a> on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)<a href="#cite_note-214">[213]</a></li> <li><a href="/wiki/Kelmans%E2%80%93Seymour_conjecture" title="Kelmans–Seymour conjecture">Kelmans–Seymour conjecture</a> (Dawei He, Yan Wang, and Xingxing Yu, 2020)<a href="#cite_note-215">[214]</a><a href="#cite_note-216">[215]</a><a href="#cite_note-217">[216]</a><a href="#cite_note-218">[217]</a></li> <li><a href="/wiki/Goldberg%E2%80%93Seymour_conjecture" title="Goldberg–Seymour conjecture">Goldberg–Seymour conjecture</a> (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)<a href="#cite_note-219">[218]</a></li> <li><a href="/wiki/Babai%27s_problem" title="Babai's problem">Babai's problem</a> (Alireza Abdollahi, Maysam Zallaghi, 2015)<a href="#cite_note-220">[219]</a></li> <li><a href="/wiki/Alspach%27s_conjecture" title="Alspach's conjecture">Alspach's conjecture</a> (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)</li> <li><a href="/wiki/Alon%E2%80%93Saks%E2%80%93Seymour_conjecture" class="mw-redirect" title="Alon–Saks–Seymour conjecture">Alon–Saks–Seymour conjecture</a> (Hao Huang, <a href="/wiki/Benny_Sudakov" title="Benny Sudakov">Benny Sudakov</a>, 2012)</li> <li><a href="/wiki/Read%27s_conjecture" title="Read's conjecture">Read–Hoggar conjecture</a> (<a href="/wiki/June_Huh" title="June Huh">June Huh</a>, 2009)<a href="#cite_note-221">[220]</a></li> <li><a href="/wiki/Scheinerman%27s_conjecture" title="Scheinerman's conjecture">Scheinerman's conjecture</a> (Jeremie Chalopin and Daniel Gonçalves, 2009)<a href="#cite_note-222">[221]</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Menger_conjecture" class="mw-redirect" title="Erdős–Menger conjecture">Erdős–Menger conjecture</a> (<a href="/wiki/Ron_Aharoni" title="Ron Aharoni">Ron Aharoni</a>, Eli Berger 2007)<a href="#cite_note-223">[222]</a></li> <li><a href="/wiki/Road_coloring_conjecture" class="mw-redirect" title="Road coloring conjecture">Road coloring conjecture</a> (<a href="/wiki/Avraham_Trahtman" title="Avraham Trahtman">Avraham Trahtman</a>, 2007)<a href="#cite_note-224">[223]</a></li> <li><a href="/wiki/Robertson%E2%80%93Seymour_theorem" title="Robertson–Seymour theorem">Robertson–Seymour theorem</a> (<a href="/wiki/Neil_Robertson_(mathematician)" title="Neil Robertson (mathematician)">Neil Robertson</a>, <a href="/wiki/Paul_Seymour_(mathematician)" title="Paul Seymour (mathematician)">Paul Seymour</a>, 2004)<a href="#cite_note-225">[224]</a></li> <li><a href="/wiki/Strong_perfect_graph_conjecture" class="mw-redirect" title="Strong perfect graph conjecture">Strong perfect graph conjecture</a> (<a href="/wiki/Maria_Chudnovsky" title="Maria Chudnovsky">Maria Chudnovsky</a>, <a href="/wiki/Neil_Robertson_(mathematician)" title="Neil Robertson (mathematician)">Neil Robertson</a>, <a href="/wiki/Paul_Seymour_(mathematician)" title="Paul Seymour (mathematician)">Paul Seymour</a> and <a href="/wiki/Robin_Thomas_(mathematician)" title="Robin Thomas (mathematician)">Robin Thomas</a>, 2002)<a href="#cite_note-226">[225]</a></li> <li><a href="/wiki/Toida%27s_conjecture" title="Toida's conjecture">Toida's conjecture</a> (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)<a href="#cite_note-227">[226]</a></li> <li>Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)<a href="#cite_note-228">[227]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Group_theory_2">Group theory</h3></div> <ul><li><a href="/wiki/Hanna_Neumann_conjecture" title="Hanna Neumann conjecture">Hanna Neumann conjecture</a> (Joel Friedman, 2011, Igor Mineyev, 2011)<a href="#cite_note-229">[228]</a><a href="#cite_note-230">[229]</a></li> <li><a href="/wiki/Density_theorem_for_Kleinian_groups" title="Density theorem for Kleinian groups">Density theorem</a> (Hossein Namazi, Juan Souto, 2010)<a href="#cite_note-231">[230]</a></li> <li>Full <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> (<a href="/wiki/Koichiro_Harada" title="Koichiro Harada">Koichiro Harada</a>, <a href="/wiki/Ronald_Solomon" title="Ronald Solomon">Ronald Solomon</a>, 2008)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Number_theory_2">Number theory</h3></div> <div class="mw-heading mw-heading4"><h4 id="21st_century_2">21st century</h4></div> <ul><li><a href="/wiki/Andr%C3%A9%E2%80%93Oort_conjecture" title="André–Oort conjecture">André–Oort conjecture</a> (<a href="/wiki/Jonathan_Pila" title="Jonathan Pila">Jonathan Pila</a>, Ananth Shankar, <a href="/wiki/Jacob_Tsimerman" title="Jacob Tsimerman">Jacob Tsimerman</a>, 2021)<a href="#cite_note-232">[231]</a></li> <li><a href="/wiki/Duffin%E2%80%93Schaeffer_conjecture" class="mw-redirect" title="Duffin–Schaeffer conjecture">Duffin-Schaeffer conjecture</a> (<a href="/wiki/Dimitris_Koukoulopoulos" title="Dimitris Koukoulopoulos">Dimitris Koukoulopoulos</a>, <a href="/wiki/James_Maynard_(mathematician)" class="mw-redirect" title="James Maynard (mathematician)">James Maynard</a>, 2019)</li> <li><a href="/wiki/Vinogradov%27s_mean-value_theorem#The_conjectured_form" title="Vinogradov's mean-value theorem">Main conjecture in Vinogradov's mean-value theorem</a> (<a href="/wiki/Jean_Bourgain" title="Jean Bourgain">Jean Bourgain</a>, Ciprian Demeter, <a href="/wiki/Larry_Guth" title="Larry Guth">Larry Guth</a>, 2015)<a href="#cite_note-233">[232]</a></li> <li><a href="/wiki/Goldbach%27s_weak_conjecture" title="Goldbach's weak conjecture">Goldbach's weak conjecture</a> (<a href="/wiki/Harald_Helfgott" title="Harald Helfgott">Harald Helfgott</a>, 2013)<a href="#cite_note-234">[233]</a><a href="#cite_note-235">[234]</a><a href="#cite_note-236">[235]</a></li> <li><a href="/wiki/Prime_gap#Further_results" title="Prime gap">Existence of bounded gaps between primes</a> (<a href="/wiki/Yitang_Zhang" title="Yitang Zhang">Yitang Zhang</a>, <a href="/wiki/Polymath_Project" title="Polymath Project">Polymath8</a>, <a href="/wiki/James_Maynard_(mathematician)" class="mw-redirect" title="James Maynard (mathematician)">James Maynard</a>, 2013)<a href="#cite_note-237">[236]</a><a href="#cite_note-238">[237]</a><a href="#cite_note-239">[238]</a></li> <li><a href="/wiki/Sidon_sequence" title="Sidon sequence">Sidon set problem</a> (Javier Cilleruelo, <a href="/wiki/Imre_Z._Ruzsa" title="Imre Z. Ruzsa">Imre Z. Ruzsa</a>, and Carlos Vinuesa, 2010)<a href="#cite_note-240">[239]</a></li> <li><a href="/wiki/Serre%27s_modularity_conjecture" title="Serre's modularity conjecture">Serre's modularity conjecture</a> (<a href="/wiki/Chandrashekhar_Khare" title="Chandrashekhar Khare">Chandrashekhar Khare</a> and <a href="/wiki/Jean-Pierre_Wintenberger" title="Jean-Pierre Wintenberger">Jean-Pierre Wintenberger</a>, 2008)<a href="#cite_note-241">[240]</a><a href="#cite_note-242">[241]</a><a href="#cite_note-243">[242]</a></li> <li><a href="/wiki/Green%E2%80%93Tao_theorem" title="Green–Tao theorem">Green–Tao theorem</a> (<a href="/wiki/Ben_J._Green" class="mw-redirect" title="Ben J. Green">Ben J. Green</a> and <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a>, 2004)<a href="#cite_note-244">[243]</a></li> <li><a href="/wiki/Mih%C4%83ilescu%27s_theorem" class="mw-redirect" title="Mihăilescu's theorem">Catalan's conjecture</a> (<a href="/wiki/Preda_Mih%C4%83ilescu" title="Preda Mihăilescu">Preda Mihăilescu</a>, 2002)<a href="#cite_note-245">[244]</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Graham_problem" title="Erdős–Graham problem">Erdős–Graham problem</a> (<a href="/wiki/Ernest_S._Croot_III" title="Ernest S. Croot III">Ernest S. Croot III</a>, 2000)<a href="#cite_note-246">[245]</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="20th_century_2">20th century</h4></div> <ul><li><a href="/wiki/Lafforgue%27s_theorem" title="Lafforgue's theorem">Lafforgue's theorem</a> (<a href="/wiki/Laurent_Lafforgue" title="Laurent Lafforgue">Laurent Lafforgue</a>, 1998)<a href="#cite_note-247">[246]</a></li> <li><a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> (<a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a> and <a href="/wiki/Richard_Taylor_(mathematician)" title="Richard Taylor (mathematician)">Richard Taylor</a>, 1995)<a href="#cite_note-248">[247]</a><a href="#cite_note-249">[248]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ramsey_theory">Ramsey theory</h3></div> <ul><li><a href="/wiki/Burr%E2%80%93Erd%C5%91s_conjecture" title="Burr–Erdős conjecture">Burr–Erdős conjecture</a> (Choongbum Lee, 2017)<a href="#cite_note-250">[249]</a></li> <li><a href="/wiki/Boolean_Pythagorean_triples_problem" title="Boolean Pythagorean triples problem">Boolean Pythagorean triples problem</a> (<a href="/wiki/Marijn_Heule" title="Marijn Heule">Marijn Heule</a>, Oliver Kullmann, <a href="/wiki/Victor_W._Marek" title="Victor W. Marek">Victor W. Marek</a>, 2016)<a href="#cite_note-251">[250]</a><a href="#cite_note-252">[251]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Theoretical_computer_science">Theoretical computer science</h3></div> <ul><li><a href="/wiki/Decision_tree_model#Sensitivity_conjecture" title="Decision tree model">Sensitivity conjecture</a> for Boolean functions (<a href="/wiki/Hao_Huang_(mathematician)" title="Hao Huang (mathematician)">Hao Huang</a>, 2019)<a href="#cite_note-253">[252]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Topology_2">Topology</h3></div> <ul><li>Deciding whether the <a href="/wiki/Conway_knot" title="Conway knot">Conway knot</a> is a <a href="/wiki/Slice_knot" title="Slice knot">slice knot</a> (<a href="/wiki/Lisa_Piccirillo" title="Lisa Piccirillo">Lisa Piccirillo</a>, 2020)<a href="#cite_note-254">[253]</a><a href="#cite_note-255">[254]</a></li> <li><a href="/wiki/Virtual_Haken_conjecture" class="mw-redirect" title="Virtual Haken conjecture">Virtual Haken conjecture</a> (<a href="/wiki/Ian_Agol" title="Ian Agol">Ian Agol</a>, Daniel Groves, Jason Manning, 2012)<a href="#cite_note-256">[255]</a> (and by work of <a href="/wiki/Daniel_Wise_(mathematician)" title="Daniel Wise (mathematician)">Daniel Wise</a> also <a href="/wiki/Virtually_fibered_conjecture" title="Virtually fibered conjecture">virtually fibered conjecture</a>)</li> <li><a href="/wiki/Hsiang%E2%80%93Lawson%27s_conjecture" title="Hsiang–Lawson's conjecture">Hsiang–Lawson's conjecture</a> (<a href="/wiki/Simon_Brendle" title="Simon Brendle">Simon Brendle</a>, 2012)<a href="#cite_note-257">[256]</a></li> <li><a href="/wiki/Ehrenpreis_conjecture" title="Ehrenpreis conjecture">Ehrenpreis conjecture</a> (<a href="/wiki/Jeremy_Kahn" title="Jeremy Kahn">Jeremy Kahn</a>, <a href="/wiki/Vladimir_Markovic" title="Vladimir Markovic">Vladimir Markovic</a>, 2011)<a href="#cite_note-258">[257]</a></li> <li><a href="/wiki/Atiyah_conjecture" title="Atiyah conjecture">Atiyah conjecture</a> for groups with finite subgroups of unbounded order (Austin, 2009)<a href="#cite_note-259">[258]</a></li> <li><a href="/wiki/Cobordism_hypothesis" title="Cobordism hypothesis">Cobordism hypothesis</a> (<a href="/wiki/Jacob_Lurie" title="Jacob Lurie">Jacob Lurie</a>, 2008)<a href="#cite_note-260">[259]</a></li> <li><a href="/wiki/Spherical_space_form_conjecture" title="Spherical space form conjecture">Spherical space form conjecture</a> (<a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>, 2006)</li> <li><a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a> (<a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>, 2002)<a href="#cite_note-auto-261">[260]</a></li> <li><a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">Geometrization conjecture</a>, (<a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>,<a href="#cite_note-auto-261">[260]</a> series of preprints in 2002–2003)<a href="#cite_note-262">[261]</a></li> <li><a href="/wiki/Nikiel%27s_conjecture" title="Nikiel's conjecture">Nikiel's conjecture</a> (<a href="/wiki/Mary_Ellen_Rudin" title="Mary Ellen Rudin">Mary Ellen Rudin</a>, 1999)<a href="#cite_note-263">[262]</a></li> <li>Disproof of the <a href="/wiki/Ganea_conjecture" title="Ganea conjecture">Ganea conjecture</a> (Iwase, 1997)<a href="#cite_note-264">[263]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Uncategorised">Uncategorised</h3></div> <div class="mw-heading mw-heading4"><h4 id="2010s">2010s</h4></div> <ul><li><a href="/wiki/Erd%C5%91s_discrepancy_problem" class="mw-redirect" title="Erdős discrepancy problem">Erdős discrepancy problem</a> (<a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a>, 2015)<a href="#cite_note-265">[264]</a></li> <li><a href="/wiki/Umbral_moonshine" title="Umbral moonshine">Umbral moonshine</a> conjecture (John F. R. Duncan, Michael J. Griffin, <a href="/wiki/Ken_Ono" title="Ken Ono">Ken Ono</a>, 2015)<a href="#cite_note-266">[265]</a></li> <li>Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (<a href="/wiki/Jeff_Cheeger" title="Jeff Cheeger">Jeff Cheeger</a>, Aaron Naber, 2014)<a href="#cite_note-267">[266]</a></li> <li><a href="/wiki/Gaussian_correlation_inequality" title="Gaussian correlation inequality">Gaussian correlation inequality</a> (<a href="/wiki/Thomas_Royen" title="Thomas Royen">Thomas Royen</a>, 2014)<a href="#cite_note-268">[267]</a></li> <li>Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, <a href="/wiki/Aleksandar_Nikolov_(computer_scientist)" title="Aleksandar Nikolov (computer scientist)">Aleksandar Nikolov</a>, 2011)<a href="#cite_note-269">[268]</a></li> <li><a href="/wiki/Bloch%E2%80%93Kato_conjecture" class="mw-redirect" title="Bloch–Kato conjecture">Bloch–Kato conjecture</a> (<a href="/wiki/Vladimir_Voevodsky" title="Vladimir Voevodsky">Vladimir Voevodsky</a>, 2011)<a href="#cite_note-270">[269]</a> (and <a href="/wiki/Quillen%E2%80%93Lichtenbaum_conjecture" title="Quillen–Lichtenbaum conjecture">Quillen–Lichtenbaum conjecture</a> and by work of <a href="/wiki/Thomas_Geisser_(mathematician)" class="mw-redirect" title="Thomas Geisser (mathematician)">Thomas Geisser</a> and <a href="/wiki/Marc_Levine_(mathematician)" title="Marc Levine (mathematician)">Marc Levine</a> (2001) also <a href="/wiki/Norm_residue_isomorphism_theorem#Beilinson–Lichtenbaum_conjecture" title="Norm residue isomorphism theorem">Beilinson–Lichtenbaum conjecture</a><a href="#cite_note-271">[270]</a><a href="#cite_note-272">[271]</a>: 359 <a href="#cite_note-273">[272]</a>)</li></ul> <div class="mw-heading mw-heading4"><h4 id="2000s">2000s</h4></div> <ul><li><a href="/w/index.php?title=Kauffman%E2%80%93Harary_conjecture&action=edit&redlink=1" class="new" title="Kauffman–Harary conjecture (page does not exist)">Kauffman–Harary conjecture</a> (Thomas Mattman, Pablo Solis, 2009)<a href="#cite_note-274">[273]</a></li> <li><a href="/wiki/Surface_subgroup_conjecture" title="Surface subgroup conjecture">Surface subgroup conjecture</a> (<a href="/wiki/Jeremy_Kahn" title="Jeremy Kahn">Jeremy Kahn</a>, <a href="/wiki/Vladimir_Markovic" title="Vladimir Markovic">Vladimir Markovic</a>, 2009)<a href="#cite_note-275">[274]</a></li> <li><a href="/w/index.php?title=Normal_scalar_curvature_conjecture&action=edit&redlink=1" class="new" title="Normal scalar curvature conjecture (page does not exist)">Normal scalar curvature conjecture</a> and the <a href="/w/index.php?title=B%C3%B6ttcher%E2%80%93Wenzel_conjecture&action=edit&redlink=1" class="new" title="Böttcher–Wenzel conjecture (page does not exist)">Böttcher–Wenzel conjecture</a> (Zhiqin Lu, 2007)<a href="#cite_note-276">[275]</a></li> <li><a href="/w/index.php?title=Nirenberg%E2%80%93Treves_conjecture&action=edit&redlink=1" class="new" title="Nirenberg–Treves conjecture (page does not exist)">Nirenberg–Treves conjecture</a> (<a href="/wiki/Nils_Dencker" title="Nils Dencker">Nils Dencker</a>, 2005)<a href="#cite_note-277">[276]</a><a href="#cite_note-278">[277]</a></li> <li><a href="/wiki/Peter_Lax" title="Peter Lax">Lax conjecture</a> (<a href="/wiki/Adrian_Lewis_(mathematician)" title="Adrian Lewis (mathematician)">Adrian Lewis</a>, <a href="/wiki/Pablo_Parrilo" title="Pablo Parrilo">Pablo Parrilo</a>, Motakuri Ramana, 2005)<a href="#cite_note-279">[278]</a></li> <li>The <a href="/wiki/Langlands%E2%80%93Shelstad_fundamental_lemma" class="mw-redirect" title="Langlands–Shelstad fundamental lemma">Langlands–Shelstad fundamental lemma</a> (<a href="/wiki/Ng%C3%B4_B%E1%BA%A3o_Ch%C3%A2u" title="Ngô Bảo Châu">Ngô Bảo Châu</a> and <a href="/wiki/G%C3%A9rard_Laumon" title="Gérard Laumon">Gérard Laumon</a>, 2004)<a href="#cite_note-280">[279]</a></li> <li><a href="/wiki/Milnor_conjecture_(K-theory)" title="Milnor conjecture (K-theory)">Milnor conjecture</a> (<a href="/wiki/Vladimir_Voevodsky" title="Vladimir Voevodsky">Vladimir Voevodsky</a>, 2003)<a href="#cite_note-281">[280]</a></li> <li><a href="/w/index.php?title=Kirillov%27s_conjecture&action=edit&redlink=1" class="new" title="Kirillov's conjecture (page does not exist)">Kirillov's conjecture</a> (Ehud Baruch, 2003)<a href="#cite_note-282">[281]</a></li> <li><a href="/w/index.php?title=Kouchnirenko%27s_conjecture&action=edit&redlink=1" class="new" title="Kouchnirenko's conjecture (page does not exist)">Kouchnirenko's conjecture</a> (Bertrand Haas, 2002)<a href="#cite_note-283">[282]</a></li> <li><a href="/wiki/N!_conjecture" title="N! conjecture">n! conjecture</a> (<a href="/wiki/Mark_Haiman" title="Mark Haiman">Mark Haiman</a>, 2001)<a href="#cite_note-284">[283]</a> (and also <a href="/wiki/Macdonald_polynomials#The_Macdonald_positivity_conjecture" title="Macdonald polynomials">Macdonald positivity conjecture</a>)</li> <li><a href="/wiki/Kato%27s_conjecture" title="Kato's conjecture">Kato's conjecture</a> (<a href="/wiki/Pascal_Auscher" title="Pascal Auscher">Pascal Auscher</a>, <a href="/wiki/Steve_Hofmann" title="Steve Hofmann">Steve Hofmann</a>, <a href="/wiki/Michael_Lacey_(mathematician)" title="Michael Lacey (mathematician)">Michael Lacey</a>, <a href="/wiki/Alan_Gaius_Ramsay_McIntosh" title="Alan Gaius Ramsay McIntosh">Alan McIntosh</a>, and Philipp Tchamitchian, 2001)<a href="#cite_note-285">[284]</a></li> <li><a href="/w/index.php?title=Deligne%27s_conjecture_on_1-motives&action=edit&redlink=1" class="new" title="Deligne's conjecture on 1-motives (page does not exist)">Deligne's conjecture on 1-motives</a> (Luca Barbieri-Viale, Andreas Rosenschon, <a href="/wiki/Morihiko_Saito" title="Morihiko Saito">Morihiko Saito</a>, 2001)<a href="#cite_note-286">[285]</a></li> <li><a href="/wiki/Modularity_theorem" title="Modularity theorem">Modularity theorem</a> (<a href="/wiki/Christophe_Breuil" title="Christophe Breuil">Christophe Breuil</a>, <a href="/wiki/Brian_Conrad" title="Brian Conrad">Brian Conrad</a>, <a href="/wiki/Fred_Diamond" title="Fred Diamond">Fred Diamond</a>, and <a href="/wiki/Richard_Taylor_(mathematician)" title="Richard Taylor (mathematician)">Richard Taylor</a>, 2001)<a href="#cite_note-287">[286]</a></li> <li><a href="/w/index.php?title=Erd%C5%91s%E2%80%93Stewart_conjecture&action=edit&redlink=1" class="new" title="Erdős–Stewart conjecture (page does not exist)">Erdős–Stewart conjecture</a> (<a href="/wiki/Florian_Luca" title="Florian Luca">Florian Luca</a>, 2001)<a href="#cite_note-288">[287]</a></li> <li><a href="/wiki/Berry%E2%80%93Robbins_problem" title="Berry–Robbins problem">Berry–Robbins problem</a> (<a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Michael Atiyah</a>, 2000)<a href="#cite_note-289">[288]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/List_of_conjectures" title="List of conjectures">List of conjectures</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_statistics" title="List of unsolved problems in statistics">List of unsolved problems in statistics</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_computer_science" title="List of unsolved problems in computer science">List of unsolved problems in computer science</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">List of unsolved problems in physics</a></li> <li><a href="/wiki/Lists_of_unsolved_problems" title="Lists of unsolved problems">Lists of unsolved problems</a></li> <li><a href="/wiki/Open_Problems_in_Mathematics" title="Open Problems in Mathematics">Open Problems in Mathematics</a></li> <li><a href="/wiki/The_Great_Mathematical_Problems" title="The Great Mathematical Problems">The Great Mathematical Problems</a></li> <li><a href="/wiki/Scottish_Book" title="Scottish Book">Scottish Book</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-163"><a href="#cite_ref-163">^</a> A disproof has been announced, with a preprint made available on <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>.<a href="#cite_note-162">[162]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output 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Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), <a rel="nofollow" class="external text" href="https://pure.uva.nl/ws/files/836396/73198_315330.pdf">"On nonobtuse simplicial partitions"</a> (PDF), SIAM Review, 51 (2): 317–335, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SIAMR..51..317B">2009SIAMR..51..317B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F060669073">10.1137/060669073</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2505583">2505583</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216078793">216078793</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181104211116/https://pure.uva.nl/ws/files/836396/73198_315330.pdf">archived</a> (PDF) from the original on 2018-11-04, retrieved 2018-11-22</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Review&rft.atitle=On+nonobtuse+simplicial+partitions&rft.volume=51&rft.issue=2&rft.pages=317-335&rft.date=2009&rft_id=info%3Adoi%2F10.1137%2F060669073&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2505583%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216078793%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2009SIAMR..51..317B&rft.aulast=Brandts&rft.aufirst=Jan&rft.au=Korotov%2C+Sergey&rft.au=K%C5%99%C3%AD%C5%BEek%2C+Michal&rft.au=%C5%A0olc%2C+Jakub&rft_id=https%3A%2F%2Fpure.uva.nl%2Fws%2Ffiles%2F836396%2F73198_315330.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988">. 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Retrieved 15 December 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=An+introduction+to+irrationality+and+transcendence+methods.&rft.date=2008&rft.aulast=Waldschmidt&rft.aufirst=Michel&rft_id=https%3A%2F%2Fwebusers.imj-prg.fr%2F~michel.waldschmidt%2Farticles%2Fpdf%2FAWSLecture5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-151"><a href="#cite_ref-151">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlbert" class="citation cs2">Albert, John, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140117150133/http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf">Some unsolved problems in number theory</a> (PDF), archived from <a rel="nofollow" class="external text" href="http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf">the original</a> (PDF) on 17 January 2014, retrieved 15 December 2014</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Some+unsolved+problems+in+number+theory&rft.aulast=Albert&rft.aufirst=John&rft_id=http%3A%2F%2Fwww2.math.ou.edu%2F~jalbert%2Fcourses%2Fopenprob2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-152"><a href="#cite_ref-152">^</a> For some background on the numbers in this problem, see articles by <a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a> at <a href="/wiki/Wolfram_MathWorld" class="mw-redirect" title="Wolfram MathWorld">Wolfram MathWorld</a> (all articles accessed 22 August 2024): <ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler's Constant</a></li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a></li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AperysConstant.html">Apéry's Constant</a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/IrrationalNumber.html">irrational numbers</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20150327024040/http://mathworld.wolfram.com/IrrationalNumber.html">Archived</a> 2015-03-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>)</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/TranscendentalNumber.html">transcendental numbers</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20141113174913/http://mathworld.wolfram.com/TranscendentalNumber.html">Archived</a> 2014-11-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>)</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/IrrationalityMeasure.html">irrationality measures</a> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20150421203736/http://mathworld.wolfram.com/IrrationalityMeasure.html">Archived</a> 2015-04-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>)</li></ul> </li> <li id="cite_note-:1-153">^ <a href="#cite_ref-:1_153-0">a</a> <a href="#cite_ref-:1_153-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaldschmidt2003" class="citation arxiv cs1">Waldschmidt, Michel (2003-12-24). 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Retrieved 2024-09-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Khinchin%27s+Constant&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FKhinchinsConstant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-156"><a href="#cite_ref-156">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAigner2013" class="citation cs2">Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-00888-2">10.1007/978-3-319-00888-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-00887-5" title="Special:BookSources/978-3-319-00887-5"><bdi>978-3-319-00887-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3098784">3098784</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov%27s+theorem+and+100+years+of+the+uniqueness+conjecture&rft.place=Cham&rft.pub=Springer&rft.date=2013&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3098784%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-319-00888-2&rft.isbn=978-3-319-00887-5&rft.aulast=Aigner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-157"><a href="#cite_ref-157">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuisman2016" class="citation arxiv cs1">Huisman, Sander G. 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Journal of the American Mathematical Society. 14 (4): 941–1006. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-01-00373-3">10.1090/S0894-0347-01-00373-3</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1839919">1839919</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9253880">9253880</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Mathematical+Society&rft.atitle=Hilbert+schemes%2C+polygraphs+and+the+Macdonald+positivity+conjecture&rft.volume=14&rft.issue=4&rft.pages=941-1006&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1839919%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9253880%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2FS0894-0347-01-00373-3&rft.aulast=Haiman&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-285"><a href="#cite_ref-285">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAuscherHofmannLaceyMcIntosh2002" class="citation journal cs1">Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan; Tchamitchian, Ph. (2002). "The solution of the Kato square root problem for second order elliptic operators on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">". Annals of Mathematics. Second Series. 156 (2): 633–654. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3597201">10.2307/3597201</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3597201">3597201</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1933726">1933726</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=The+solution+of+the+Kato+square+root+problem+for+second+order+elliptic+operators+on+MATH+RENDER+ERROR&rft.volume=156&rft.issue=2&rft.pages=633-654&rft.date=2002&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1933726%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3597201%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3597201&rft.aulast=Auscher&rft.aufirst=Pascal&rft.au=Hofmann%2C+Steve&rft.au=Lacey%2C+Michael&rft.au=McIntosh%2C+Alan&rft.au=Tchamitchian%2C+Ph.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-286"><a href="#cite_ref-286">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarbieri-VialeRosenschonSaito2003" class="citation journal cs1">Barbieri-Viale, Luca; Rosenschon, Andreas; Saito, Morihiko (2003). <a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2003.158.593">"Deligne's Conjecture on 1-Motives"</a>. Annals of Mathematics. 158 (2): 593–633. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0102150">math/0102150</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2003.158.593">10.4007/annals.2003.158.593</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Deligne%27s+Conjecture+on+1-Motives&rft.volume=158&rft.issue=2&rft.pages=593-633&rft.date=2003&rft_id=info%3Aarxiv%2Fmath%2F0102150&rft_id=info%3Adoi%2F10.4007%2Fannals.2003.158.593&rft.aulast=Barbieri-Viale&rft.aufirst=Luca&rft.au=Rosenschon%2C+Andreas&rft.au=Saito%2C+Morihiko&rft_id=https%3A%2F%2Fdoi.org%2F10.4007%252Fannals.2003.158.593&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-287"><a href="#cite_ref-287">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBreuilConradDiamondTaylor2001" class="citation cs2">Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", <a href="/wiki/Journal_of_the_American_Mathematical_Society" title="Journal of the American Mathematical Society">Journal of the American Mathematical Society</a>, 14 (4): 843–939, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-01-00370-8">10.1090/S0894-0347-01-00370-8</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0894-0347">0894-0347</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1839918">1839918</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Mathematical+Society&rft.atitle=On+the+modularity+of+elliptic+curves+over+Q%3A+wild+3-adic+exercises&rft.volume=14&rft.issue=4&rft.pages=843-939&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1839918%23id-name%3DMR&rft.issn=0894-0347&rft_id=info%3Adoi%2F10.1090%2FS0894-0347-01-00370-8&rft.aulast=Breuil&rft.aufirst=Christophe&rft.au=Conrad%2C+Brian&rft.au=Diamond%2C+Fred&rft.au=Taylor%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-288"><a href="#cite_ref-288">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuca2000" class="citation journal cs1">Luca, Florian (2000). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf">"On a conjecture of Erdős and Stewart"</a> (PDF). Mathematics of Computation. 70 (234): 893–897. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001MaCom..70..893L">2001MaCom..70..893L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-00-01178-9">10.1090/s0025-5718-00-01178-9</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160402030443/http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf">Archived</a> (PDF) from the original on 2016-04-02. Retrieved 2016-03-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=On+a+conjecture+of+Erd%C5%91s+and+Stewart&rft.volume=70&rft.issue=234&rft.pages=893-897&rft.date=2000&rft_id=info%3Adoi%2F10.1090%2Fs0025-5718-00-01178-9&rft_id=info%3Abibcode%2F2001MaCom..70..893L&rft.aulast=Luca&rft.aufirst=Florian&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F2001-70-234%2FS0025-5718-00-01178-9%2FS0025-5718-00-01178-9.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> <li id="cite_note-289"><a href="#cite_ref-289">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtiyah2000" class="citation book cs1"><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael</a> (2000). "The geometry of classical particles". In <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau, Shing-Tung</a> (ed.). Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer. Surveys in Differential Geometry. Vol. 7. Somerville, Massachusetts: International Press. pp. 1–15. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FSDG.2002.v7.n1.a1">10.4310/SDG.2002.v7.n1.a1</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1919420">1919420</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+geometry+of+classical+particles&rft.btitle=Papers+dedicated+to+Atiyah%2C+Bott%2C+Hirzebruch%2C+and+Singer&rft.place=Somerville%2C+Massachusetts&rft.series=Surveys+in+Differential+Geometry&rft.pages=1-15&rft.pub=International+Press&rft.date=2000&rft_id=info%3Adoi%2F10.4310%2FSDG.2002.v7.n1.a1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1919420%23id-name%3DMR&rft.aulast=Atiyah&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <div class="mw-heading mw-heading3"><h3 id="Books_discussing_problems_solved_since_1995">Books discussing problems solved since 1995</h3></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingh2002" class="citation book cs1"><a href="/wiki/Simon_Singh" title="Simon Singh">Singh, Simon</a> (2002). <a href="/wiki/Fermat%27s_Last_Theorem_(book)" title="Fermat's Last Theorem (book)">Fermat's Last Theorem</a>. Fourth Estate. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84115-791-7" title="Special:BookSources/978-1-84115-791-7"><bdi>978-1-84115-791-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fermat%27s+Last+Theorem&rft.pub=Fourth+Estate&rft.date=2002&rft.isbn=978-1-84115-791-7&rft.aulast=Singh&rft.aufirst=Simon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Shea2007" class="citation book cs1"><a href="/wiki/Donal_O%27Shea" title="Donal O'Shea">O'Shea, Donal</a> (2007). The Poincaré Conjecture. Penguin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84614-012-9" title="Special:BookSources/978-1-84614-012-9"><bdi>978-1-84614-012-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Poincar%C3%A9+Conjecture&rft.pub=Penguin&rft.date=2007&rft.isbn=978-1-84614-012-9&rft.aulast=O%27Shea&rft.aufirst=Donal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzpiro2003" class="citation book cs1"><a href="/wiki/George_Szpiro" title="George Szpiro">Szpiro, George G.</a> (2003). Kepler's Conjecture. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-08601-7" title="Special:BookSources/978-0-471-08601-7"><bdi>978-0-471-08601-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kepler%27s+Conjecture&rft.pub=Wiley&rft.date=2003&rft.isbn=978-0-471-08601-7&rft.aulast=Szpiro&rft.aufirst=George+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRonan2006" class="citation book cs1"><a href="/wiki/Mark_Ronan" title="Mark Ronan">Ronan, Mark</a> (2006). Symmetry and the Monster. Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-280722-9" title="Special:BookSources/978-0-19-280722-9"><bdi>978-0-19-280722-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+and+the+Monster&rft.pub=Oxford&rft.date=2006&rft.isbn=978-0-19-280722-9&rft.aulast=Ronan&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Books_discussing_unsolved_problems">Books discussing unsolved problems</h3></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChungGraham1999" class="citation book cs1"><a href="/wiki/Fan_Chung" title="Fan Chung">Chung, Fan</a>; <a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ron</a> (1999). <a href="/wiki/Erd%C5%91s_on_Graphs" title="Erdős on Graphs">Erdös on Graphs: His Legacy of Unsolved Problems</a>. AK Peters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-111-6" title="Special:BookSources/978-1-56881-111-6"><bdi>978-1-56881-111-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Erd%C3%B6s+on+Graphs%3A+His+Legacy+of+Unsolved+Problems&rft.pub=AK+Peters&rft.date=1999&rft.isbn=978-1-56881-111-6&rft.aulast=Chung&rft.aufirst=Fan&rft.au=Graham%2C+Ron&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCroftFalconerGuy1994" class="citation book cs1">Croft, Hallard T.; <a href="/wiki/Kenneth_Falconer_(mathematician)" title="Kenneth Falconer (mathematician)">Falconer, Kenneth J.</a>; <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/unsolvedproblems0000crof">Unsolved Problems in Geometry</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97506-1" title="Special:BookSources/978-0-387-97506-1"><bdi>978-0-387-97506-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Geometry&rft.pub=Springer&rft.date=1994&rft.isbn=978-0-387-97506-1&rft.aulast=Croft&rft.aufirst=Hallard+T.&rft.au=Falconer%2C+Kenneth+J.&rft.au=Guy%2C+Richard+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funsolvedproblems0000crof&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation book cs1"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004). Unsolved Problems in Number Theory. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20860-2" title="Special:BookSources/978-0-387-20860-2"><bdi>978-0-387-20860-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.pub=Springer&rft.date=2004&rft.isbn=978-0-387-20860-2&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleeWagon1996" class="citation book cs1"><a href="/wiki/Victor_Klee" title="Victor Klee">Klee, Victor</a>; <a href="/wiki/Stan_Wagon" title="Stan Wagon">Wagon, Stan</a> (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/oldnewunsolvedpr0000klee">Old and New Unsolved Problems in Plane Geometry and Number Theory</a>. The Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-315-3" title="Special:BookSources/978-0-88385-315-3"><bdi>978-0-88385-315-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Old+and+New+Unsolved+Problems+in+Plane+Geometry+and+Number+Theory&rft.pub=The+Mathematical+Association+of+America&rft.date=1996&rft.isbn=978-0-88385-315-3&rft.aulast=Klee&rft.aufirst=Victor&rft.au=Wagon%2C+Stan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foldnewunsolvedpr0000klee&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFdu_Sautoy2003" class="citation book cs1"><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">du Sautoy, Marcus</a> (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/musicofprimes00marc">The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics</a>. Harper Collins. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-06-093558-0" title="Special:BookSources/978-0-06-093558-0"><bdi>978-0-06-093558-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Music+of+the+Primes%3A+Searching+to+Solve+the+Greatest+Mystery+in+Mathematics&rft.pub=Harper+Collins&rft.date=2003&rft.isbn=978-0-06-093558-0&rft.aulast=du+Sautoy&rft.aufirst=Marcus&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmusicofprimes00marc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDerbyshire2003" class="citation book cs1"><a href="/wiki/John_Derbyshire" title="John Derbyshire">Derbyshire, John</a> (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/primeobsessionbe00derb_0">Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics</a>. Joseph Henry Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-309-08549-6" title="Special:BookSources/978-0-309-08549-6"><bdi>978-0-309-08549-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prime+Obsession%3A+Bernhard+Riemann+and+the+Greatest+Unsolved+Problem+in+Mathematics&rft.pub=Joseph+Henry+Press&rft.date=2003&rft.isbn=978-0-309-08549-6&rft.aulast=Derbyshire&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprimeobsessionbe00derb_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin2006" class="citation book cs1"><a href="/wiki/Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7607-8659-8" title="Special:BookSources/978-0-7607-8659-8"><bdi>978-0-7607-8659-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Millennium+Problems+%E2%80%93+The+Seven+Greatest+Unsolved%2A+Mathematical+Puzzles+Of+Our+Time&rft.pub=Barnes+%26+Noble&rft.date=2006&rft.isbn=978-0-7607-8659-8&rft.aulast=Devlin&rft.aufirst=Keith&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlondelMegrestski2004" class="citation book cs1"><a href="/wiki/Vincent_Blondel" title="Vincent Blondel">Blondel, Vincent D.</a>; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11748-5" title="Special:BookSources/978-0-691-11748-5"><bdi>978-0-691-11748-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+problems+in+mathematical+systems+and+control+theory&rft.pub=Princeton+University+Press&rft.date=2004&rft.isbn=978-0-691-11748-5&rft.aulast=Blondel&rft.aufirst=Vincent+D.&rft.au=Megrestski%2C+Alexandre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiPoonYau2013" class="citation book cs1"><a href="/wiki/Lizhen_Ji" title="Lizhen Ji">Ji, Lizhen</a>; Poon, Yat-Sun; <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau, Shing-Tung</a> (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57146-278-7" title="Special:BookSources/978-1-57146-278-7"><bdi>978-1-57146-278-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Open+Problems+and+Surveys+of+Contemporary+Mathematics+%28volume+6+in+the+Surveys+in+Modern+Mathematics+series%29+%28Surveys+of+Modern+Mathematics%29&rft.pub=International+Press+of+Boston&rft.date=2013&rft.isbn=978-1-57146-278-7&rft.aulast=Ji&rft.aufirst=Lizhen&rft.au=Poon%2C+Yat-Sun&rft.au=Yau%2C+Shing-Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaldschmidt2004" class="citation journal cs1"><a href="/wiki/Michel_Waldschmidt" title="Michel Waldschmidt">Waldschmidt, Michel</a> (2004). <a rel="nofollow" class="external text" href="http://www.math.jussieu.fr/~miw/articles/pdf/odp.pdf">"Open Diophantine Problems"</a> (PDF). Moscow Mathematical Journal. 4 (1): 245–305. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0312440">math/0312440</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17323%2F1609-4514-2004-4-1-245-305">10.17323/1609-4514-2004-4-1-245-305</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1609-3321">1609-3321</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11845578">11845578</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1066.11030">1066.11030</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Moscow+Mathematical+Journal&rft.atitle=Open+Diophantine+Problems&rft.volume=4&rft.issue=1&rft.pages=245-305&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1066.11030%23id-name%3DZbl&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11845578%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.17323%2F1609-4514-2004-4-1-245-305&rft_id=info%3Aarxiv%2Fmath%2F0312440&rft.issn=1609-3321&rft.aulast=Waldschmidt&rft.aufirst=Michel&rft_id=http%3A%2F%2Fwww.math.jussieu.fr%2F~miw%2Farticles%2Fpdf%2Fodp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMazurovKhukhro2015" class="citation arxiv cs1"><a href="/wiki/Victor_Mazurov" title="Victor Mazurov">Mazurov, V. D.</a>; Khukhro, E. I. (1 Jun 2015). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1401.0300v6">1401.0300v6</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GR">math.GR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Unsolved+Problems+in+Group+Theory.+The+Kourovka+Notebook.+No.+18+%28English+version%29&rft.date=2015-06-01&rft_id=info%3Aarxiv%2F1401.0300v6&rft.aulast=Mazurov&rft.aufirst=V.+D.&rft.au=Khukhro%2C+E.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="http://faculty.evansville.edu/ck6/integer/unsolved.html">24 Unsolved Problems and Rewards for them</a></li> <li><a rel="nofollow" class="external text" href="http://www.openproblems.net/">List of links to unsolved problems in mathematics, prizes and research</a></li> <li><a rel="nofollow" class="external text" href="http://garden.irmacs.sfu.ca/">Open Problem Garden</a></li> <li><a rel="nofollow" class="external text" href="http://aimpl.org/">AIM Problem Lists</a></li> <li><a rel="nofollow" class="external text" href="http://cage.ugent.be/~hvernaev/problems/archive.html">Unsolved Problem of the Week Archive</a>. MathPro Press.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall" class="citation web cs1"><a href="/wiki/John_M._Ball" title="John M. Ball">Ball, John M.</a> <a rel="nofollow" class="external text" href="https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf">"Some Open Problems in Elasticity"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Some+Open+Problems+in+Elasticity&rft.aulast=Ball&rft.aufirst=John+M.&rft_id=https%3A%2F%2Fpeople.maths.ox.ac.uk%2Fball%2FArticles%2520in%2520Conference%2520Proceedings%2520and%2520Books%2FJMB%25202002%2520re%2520Marsden%252060th.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConstantin" class="citation web cs1"><a href="/w/index.php?title=Peter_Constantin&action=edit&redlink=1" class="new" title="Peter Constantin (page does not exist)">Constantin, Peter</a>. <a rel="nofollow" class="external text" href="https://web.math.princeton.edu/~const/2k.pdf">"Some open problems and research directions in the mathematical study of fluid dynamics"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Some+open+problems+and+research+directions+in+the+mathematical+study+of+fluid+dynamics&rft.aulast=Constantin&rft.aufirst=Peter&rft_id=https%3A%2F%2Fweb.math.princeton.edu%2F~const%2F2k.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre" class="citation web cs1"><a href="/wiki/Denis_Serre" title="Denis Serre">Serre, Denis</a>. <a rel="nofollow" class="external text" href="http://www.umpa.ens-lyon.fr/~serre/DPF/Ouverts.pdf">"Five Open Problems in Compressible Mathematical Fluid Dynamics"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Five+Open+Problems+in+Compressible+Mathematical+Fluid+Dynamics&rft.aulast=Serre&rft.aufirst=Denis&rft_id=http%3A%2F%2Fwww.umpa.ens-lyon.fr%2F~serre%2FDPF%2FOuverts.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://unsolvedproblems.org/">Unsolved Problems in Number Theory, Logic and Cryptography</a></li> <li><a rel="nofollow" class="external text" href="http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html">200 open problems in graph theory</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170515145908/http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html">Archived</a> 2017-05-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://cs.smith.edu/~orourke/TOPP/">The Open Problems Project (TOPP)</a>, discrete and computational geometry problems</li> <li><a rel="nofollow" class="external text" href="http://math.berkeley.edu/~kirby/problems.ps.gz">Kirby's list of unsolved problems in low-dimensional topology</a></li> <li><a rel="nofollow" class="external text" href="http://www.math.ucsd.edu/~erdosproblems/">Erdös' Problems on Graphs</a></li> <li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1409.2823">Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory</a></li> <li><a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/S0165011414003194">Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications</a></li> <li><a rel="nofollow" class="external text" href="http://wwwmath.uni-muenster.de/logik/Personen/rds/list.html">List of open problems in inner model theory</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAizenman" class="citation web cs1"><a href="/wiki/Michael_Aizenman" title="Michael Aizenman">Aizenman, Michael</a>. <a rel="nofollow" class="external text" href="https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/OPlist.html">"Open Problems in Mathematical Physics"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Open+Problems+in+Mathematical+Physics&rft.aulast=Aizenman&rft.aufirst=Michael&rft_id=https%3A%2F%2Fweb.math.princeton.edu%2F~aizenman%2FOpenProblems_MathPhys%2FOPlist.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+unsolved+problems+in+mathematics" class="Z3988"></li> <li><a href="/wiki/Barry_Simon" title="Barry Simon">Barry Simon</a>'s <a rel="nofollow" class="external text" href="http://math.caltech.edu/SimonPapers/R27.pdf">15 Problems in Mathematical Physics</a></li> <li><a href="/wiki/Alexandre_Eremenko" title="Alexandre Eremenko">Alexandre Eremenko</a>. <a rel="nofollow" class="external text" href="https://www.math.purdue.edu/~eremenko/uns1.html">Unsolved problems in Function Theory</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Well-known_unsolved_problems_by_discipline" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a 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