Document Zbl 0071.03702

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It is conjectured that the only other solutions have $m = n = 1$, apart from $p = q = 2$, $n = 24$ and $m = 70$. This conjecture is verified for a large number of values of $p$ and $q$.<br class="zbmathjax-paragraph">It is shown that the problem of solving (1) may be reduced to the problem of solving a finite number of equations of the form \[ | Ax^q - By^q| = 1, \tag{2} \] where $A$ and $B$ are bounded by $p$ alone. By a theorem of Nagell, Ljunggren and <span class="zbmathjax-textit">Y. Domar</span> [Math. Scand. 2, 29–32 (1954; <a href="/0056.03602">Zbl 0056.03602</a>)] each such equation has at most two solutions in positive integers $x$ and $y$ for $q\ge 3$. The equation (2) is further shown insoluble for wide classes of values of $p$ and $q$, where the author now makes use of a large number of known theorems concerning various diophantine equations. <br class="zbmathjax-paragraph">The reviewer remarks that the proof of Lucas for the case $p = q = 2$ is insufficient. The first complete solution was given by <span class="zbmathjax-textit">G. N. Watson</span> [Mess. Math. 48 (1918–1919), 1–22 (1919; <a href="/46.0213.01">JFM 46.0213.01</a>)].<div class="reviewer"> Reviewer: <a href="/authors/?q=W%2A+Ljunggren">W. Ljunggren</a></div> <div class="clearfix"></div><div class="scan"> <span> Page: <a class="goto" href="/scans/071/032.gif">−5</a> <a class="goto" href="/scans/071/033.gif">−4</a> <a class="goto" href="/scans/071/034.gif">−3</a> <a class="goto" href="/scans/071/035.gif">−2</a> <a class="goto" href="/scans/071/036.gif">−1</a> <a class="goto" href="/scans/071/037.gif">±0</a> <a class="goto" href="/scans/071/038.gif">+1</a> <a class="goto" href="/scans/071/039.gif">+2</a> <a class="goto" href="/scans/071/040.gif">+3</a> <a class="goto" href="/scans/071/041.gif">+4</a> <a class="goto" href="/scans/071/042.gif">+5</a></span> <a class="scan" href="/scans/071/037.gif">Show Scanned Page</a><br> <img src="/scans/071/037.gif" alt="scan of Zbl 0071.03702" onclick="$(this).siblings('.scan').click();"> </div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3118517">Cited in <strong>5</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A3118517">Cited in <strong>19</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A11D61" title="MSC2020">11D61</a> </td> <td class="space"> Exponential Diophantine equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aexponential+Diophantine+equations">exponential Diophantine equations</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0056.03602">Zbl 0056.03602</a>; <a href="/46.0213.01">JFM 46.0213.01</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0071.03702" data-ciurl="/ci/03118517" data-biburl="/bibtex/03118517.bib" data-amsurl="/amsrefs/03118517.bib" data-xmlurl="/xml/03118517.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03118517.pdf" title="Zbl 0071.03702 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF02401100" aria-label="DOI for “The equation $1^p+2^p+3^p+\ldots+n^p=m^q$”" title="10.1007/BF02401100">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">N. G. Ankeny andP. Erdös, The insolubility of classes of diophantine equations.Am. Journ. Math., 76 (1954), 488–496. · <a href="/0056.03505" class="nowrap">Zbl 0056.03505</a> · <a href="https://doi.org/10.2307/2372696" class="nowrap">doi:10.2307/2372696</a></td> </tr><tr> <td>[2]</td> <td class="space">Appell etGoursat,Théorie des fonctions algébriques. Paris 1929.</td> </tr><tr> <td>[3]</td> <td class="space">N. G. W. H. Beeger, On the congruence 2 p (modp 2) and Fermat’s Last Theorem.Nieuw Arch. Wiskunde, 20 (1939), 51–54. · <a href="/0020.10505" class="nowrap">Zbl 0020.10505</a></td> </tr><tr> <td>[4]</td> <td class="space">J. W. S. Cassels, On the equationa x /t-b y =1.Am. Journ. Math. 75 (1953), 159–162. · <a href="/0050.03703" class="nowrap">Zbl 0050.03703</a> · <a href="https://doi.org/10.2307/2372624" class="nowrap">doi:10.2307/2372624</a></td> </tr><tr> <td>[5]</td> <td class="space">P. Dénes, An extension of Legendre’s criterion in connection with the first case of Fermat’s last theorem.Publ. Math. Debrecen, 2 (1951), 115–120. · <a href="/0043.27302" class="nowrap">Zbl 0043.27302</a></td> </tr><tr> <td>[6]</td> <td class="space">–, Über die diophantische Gleichungx l +y l =cz l .Acta Math., 88 (1952), 241–251. · <a href="/0048.27503" class="nowrap">Zbl 0048.27503</a> · <a href="https://doi.org/10.1007/BF02392133" class="nowrap">doi:10.1007/BF02392133</a></td> </tr><tr> <td>[7]</td> <td class="space">L. E. Dickson,History of the Theory of Numbers, Vol. II. · <a href="/48.0137.02" class="nowrap">JFM 48.0137.02</a></td> </tr><tr> <td>[8]</td> <td class="space">L. E. Dickson,Introduction to the Theory of Numbers. 1929. · <a href="/55.0092.19" class="nowrap">JFM 55.0092.19</a></td> </tr><tr> <td>[9]</td> <td class="space">Y. Domar, On the diophantine equation /vbAx n /t-By n /vb=1.Math. Scand., 2 (1954), 29–32. · <a href="/0056.03602" class="nowrap">Zbl 0056.03602</a></td> </tr><tr> <td>[10]</td> <td class="space">M. Gérono, Solution de la Question 1177.Nouv. Ann. Math. (2), 16 (1877), 230–234.</td> </tr><tr> <td>[11]</td> <td class="space">E. Landau,Vorlesungen über Zahlentheorie. 1927. · <a href="/53.0123.17" class="nowrap">JFM 53.0123.17</a></td> </tr><tr> <td>[12]</td> <td class="space">V. A. Lebesgue, Théorèmes nouveaux sur l’équation indéterminéex 5 +y 5 =az 5 . Journal de Math. (1), 8 (1843), 49–70.</td> </tr><tr> <td>[13]</td> <td class="space">A. M. Legendre,Théorie des Nombres, IV, No 1. 1830.</td> </tr><tr> <td>[14]</td> <td class="space">D. H. Lehmer, E. Lehmer andH. S. Vandiver, An application of high-speed computing to Fermat’s Last Theorem.Proc. Nat. Acad. Sci. USA, 40 (1954), 25–33. · <a href="/0055.04004" class="nowrap">Zbl 0055.04004</a> · <a href="https://doi.org/10.1073/pnas.40.1.25" class="nowrap">doi:10.1073/pnas.40.1.25</a></td> </tr><tr> <td>[15]</td> <td class="space">J. Liouville, Sur l’équationZ 2n 2n =2x n .Journal de Math. (1), 5 (1840), 360.</td> </tr><tr> <td>[16]</td> <td class="space">W. Ljunggren, Über die Lösung einiger unbestimmten Gleichungen vierten Grades.Avhdl. Norske Vid. Akad. Oslo (1935), 1–35. · <a href="/61.0179.02" class="nowrap">JFM 61.0179.02</a></td> </tr><tr> <td>[17]</td> <td class="space">W. Ljunggren, Zur Theorie der Gleichungx 2 +1=Dy 4 .Avhdl. Norske Vid. Akad. Oslo. I 5 (1942). · <a href="/0027.01103" class="nowrap">Zbl 0027.01103</a></td> </tr><tr> <td>[18]</td> <td class="space">–, Über die Gleichungen1+Dx 2 =2y n ,1+Dx 2 =4y n .Norske Vid. Selsk., Forhdl., 15 (1953), 115–118.</td> </tr><tr> <td>[19]</td> <td class="space">–, New solution of a problem proposed by E. Lucas.Norsk Math. Tidskr. 34 (1952), 65–72. · <a href="/0047.04102" class="nowrap">Zbl 0047.04102</a></td> </tr><tr> <td>[20]</td> <td class="space">É. Lucas, Sur la résolution du système des équationsx 2 /t-6y 2 =u 2 ,x 2 +6y 2 =v 2 .Nouv. Ann. Math., (2), 15 (1876), 466–470.</td> </tr><tr> <td>[21]</td> <td class="space">–, Sur la résolution du système des équations2v 2 /t-u 2 =w 2 et2v 2 +u 2 =3z 2 en nombres entiers.Nouv. Ann. Math. (2), 16 (1877), 409–416.</td> </tr><tr> <td>[22]</td> <td class="space">É. Lucas, Solution de la Question 1180.Nouv. Ann. Math. (2), 16 (1877), 429–432.</td> </tr><tr> <td>[23]</td> <td class="space">T. Nagell, Über die Einheiten in reinen kubischen Zahlkörpern.Skrifter Vid. Selsk. Christiania (1922).</td> </tr><tr> <td>[24]</td> <td class="space">–, Solution complète de quelques équations cubiques à deux indéterminées.Journal de Math. (9), 4 (1925), 209–270. · <a href="/51.0135.02" class="nowrap">JFM 51.0135.02</a></td> </tr><tr> <td>[25]</td> <td class="space">N. E. Nörlund, Mémoire sur les polynômes de Bernoulli.Acta Math., 43 (1922), 121–196. · <a href="/47.0216.05" class="nowrap">JFM 47.0216.05</a> · <a href="https://doi.org/10.1007/BF02401755" class="nowrap">doi:10.1007/BF02401755</a></td> </tr><tr> <td>[26]</td> <td class="space">E. S. Selmer, The diophantine equationax 3 +by 3 +cz 3 =0.Acta Math., 85 (1951), 203–362. · <a href="/0042.26905" class="nowrap">Zbl 0042.26905</a> · <a href="https://doi.org/10.1007/BF02395746" class="nowrap">doi:10.1007/BF02395746</a></td> </tr><tr> <td>[27]</td> <td class="space">C. L. Siegel, Über einige Anwendungen diophantischer Approximationen.Abh. preuss. Akad. Wiss., phys.-math. Kl. 1929, No 1 1–70.</td> </tr><tr> <td>[28]</td> <td class="space">J. V. Uspensky andM. A. Heaslet.Elementary Number Theory. 1939. · <a href="/0022.30602" class="nowrap">Zbl 0022.30602</a></td> </tr><tr> <td>[29]</td> <td class="space">B. L. van der Waerden,Modern Algebra. 1949.</td> </tr><tr> <td>[30]</td> <td class="space">H. S. Vandiver, On classes of diophantine equations of higher degrees which have no solutions.Proc. Nat. Acad. Sci. USA., 32 (1946), 101–106. · <a href="/0063.07966" class="nowrap">Zbl 0063.07966</a> · <a href="https://doi.org/10.1073/pnas.32.4.101" class="nowrap">doi:10.1073/pnas.32.4.101</a></td> </tr><tr> <td>[31]</td> <td class="space">”X”. Letter to L. J. Mordell,Journal London Math. Soc., I (1926), 66–68. · <a href="/52.0149.02" class="nowrap">JFM 52.0149.02</a> · <a href="https://doi.org/10.1112/jlms/s1-1.2.66" class="nowrap">doi:10.1112/jlms/s1-1.2.66</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2024 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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