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id="mf-section-0"> An Egyptian fraction is a finite sum of distinct <a href="/wiki/Unit_fraction" title="Unit fraction">unit fractions</a>, such as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06c170f3a45f7e8d6733961e7a493ea7f52ccfa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.486ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.}"> That is, each <a href="/wiki/Fraction" title="Fraction">fraction</a> in the expression has a <a href="/wiki/Numerator" class="mw-redirect" title="Numerator">numerator</a> equal to 1 and a <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a> that is a positive <a href="/wiki/Integer" title="Integer">integer</a>, and all the denominators differ from each other. The value of an expression of this type is a <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a> <a href="/wiki/Rational_number" title="Rational number">rational number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e9c32a14514b5b975a4666af015884bc93b0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{b}}}">; for instance the Egyptian fraction above sums to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {43}{48}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>43</mn> <mn>48</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {43}{48}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c380f0e524d1f7063ddf0767d1a88a23d8a6f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.843ex;" alt="{\displaystyle {\tfrac {43}{48}}}">. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}"> as <a href="/wiki/Summand" class="mw-redirect" title="Summand">summands</a>, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by <a href="/wiki/Vulgar_fraction" class="mw-redirect" title="Vulgar fraction">vulgar fractions</a> and <a href="/wiki/Decimal" title="Decimal">decimal</a> notation. However, Egyptian fractions continue to be an object of study in modern <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>, as well as in modern historical studies of <a href="/wiki/History_of_mathematics" title="History of mathematics">ancient mathematics</a>. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Rhind_Mathematical_Papyrus.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/300px-Rhind_Mathematical_Papyrus.jpg" decoding="async" width="300" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/450px-Rhind_Mathematical_Papyrus.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Rhind_Mathematical_Papyrus.jpg/600px-Rhind_Mathematical_Papyrus.jpg 2x" data-file-width="750" data-file-height="449"></a><figcaption>The <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a></figcaption></figure> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Applications">1 Applications</a></li> <li class="toclevel-1 tocsection-2"><a href="#Early_history">2 Early history</a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Notation">2.1 Notation</a></li> <li class="toclevel-2 tocsection-4"><a href="#Calculation_methods">2.2 Calculation methods</a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Later_usage">3 Later usage</a></li> <li class="toclevel-1 tocsection-6"><a href="#Modern_number_theory">4 Modern number theory</a></li> <li class="toclevel-1 tocsection-7"><a href="#Open_problems">5 Open problems</a></li> <li class="toclevel-1 tocsection-8"><a href="#See_also">6 See also</a></li> <li class="toclevel-1 tocsection-9"><a href="#Notes">7 Notes</a></li> <li class="toclevel-1 tocsection-10"><a href="#References">8 References</a></li> <li class="toclevel-1 tocsection-11"><a href="#External_links">9 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Applications">Applications</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=1" title="Edit section: Applications" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> Beyond their historical use, Egyptian fractions have some practical advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing food or other objects into equal shares.<a href="#cite_note-1">[1]</a> For example, if one wants to divide 5 pizzas equally among 8 diners, the Egyptian fraction <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5}{8}}={\frac {1}{2}}+{\frac {1}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{8}}={\frac {1}{2}}+{\frac {1}{8}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db77695119e9c0d81bfb86b20bf0b9989994170c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.935ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{8}}={\frac {1}{2}}+{\frac {1}{8}}}"></noscript><span class="lazy-image-placeholder" style="width: 11.935ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db77695119e9c0d81bfb86b20bf0b9989994170c" data-alt="{\displaystyle {\frac {5}{8}}={\frac {1}{2}}+{\frac {1}{8}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  means that each diner gets half a pizza plus another eighth of a pizza, for example by splitting 4 pizzas into 8 halves, and the remaining pizza into 8 eighths. Exercises in performing this sort of <a href="/wiki/Fair_division" title="Fair division">fair division</a> of food are a standard classroom example in teaching students to work with unit fractions.<a href="#cite_note-FOOTNOTEWilsonEdgingtonNguyenPescosolido2011-2">[2]</a> Egyptian fractions can provide a solution to <a href="/wiki/Rope-burning_puzzle" title="Rope-burning puzzle">rope-burning puzzles</a>, in which a given duration is to be measured by igniting non-uniform ropes which burn out after a unit time. Any rational fraction of a unit of time can be measured by expanding the fraction into a sum of unit fractions and then, for each unit fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" 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 1/x}"></noscript><span class="lazy-image-placeholder" style="width: 3.655ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" data-alt="{\displaystyle 1/x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , burning a rope so that it always has <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  simultaneously lit points where it is burning. For this application, it is not necessary for the unit fractions to be distinct from each other. However, this solution may need an infinite number of re-lighting steps.<a href="#cite_note-FOOTNOTEWinkler2004-3">[3]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Early_history">Early history</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=2" title="Edit section: Early history" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian numerals</a> and <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a></div> Egyptian fraction notation was developed in the <a href="/wiki/Middle_Kingdom_of_Egypt" title="Middle Kingdom of Egypt">Middle Kingdom of Egypt</a>. Five early texts in which Egyptian fractions appear were the <a href="/wiki/Egyptian_Mathematical_Leather_Roll" title="Egyptian Mathematical Leather Roll">Egyptian Mathematical Leather Roll</a>, the <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow Mathematical Papyrus</a>, the <a href="/wiki/Reisner_Papyrus" title="Reisner Papyrus">Reisner Papyrus</a>, the <a href="/wiki/Kahun_Papyrus" class="mw-redirect" title="Kahun Papyrus">Kahun Papyrus</a> and the <a href="/wiki/Akhmim_Wooden_Tablet" class="mw-redirect" title="Akhmim Wooden Tablet">Akhmim Wooden Tablet</a>. A later text, the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a>, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by <a href="/wiki/Ahmes" title="Ahmes">Ahmes</a> and dates from the <a href="/wiki/Second_Intermediate_Period" class="mw-redirect" title="Second Intermediate Period">Second Intermediate Period</a>; it includes a <a href="/wiki/RMP_2/n_table" class="mw-redirect" title="RMP 2/n table">table of Egyptian fraction expansions for rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{n}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada21e7887bd500f615e37467aa9a0780c48a983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{n}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.822ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada21e7887bd500f615e37467aa9a0780c48a983" data-alt="{\displaystyle {\tfrac {2}{n}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </a>, as well as 84 <a href="/wiki/Word_problem_(mathematics_education)" title="Word problem (mathematics education)">word problems</a>. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. Tables of expansions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{n}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada21e7887bd500f615e37467aa9a0780c48a983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{n}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.822ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada21e7887bd500f615e37467aa9a0780c48a983" data-alt="{\displaystyle {\tfrac {2}{n}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the <a href="/wiki/Kahun_Papyrus" class="mw-redirect" title="Kahun Papyrus">Kahun Papyrus</a> shows, <a href="/wiki/Vulgar_fraction" class="mw-redirect" title="Vulgar fraction">vulgar fractions</a> were also used by scribes within their calculations. <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=3" title="Edit section: Notation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the <a href="/wiki/Egyptian_hieroglyphs" title="Egyptian hieroglyphs">hieroglyph</a>: <table border="0" style="margin-left: 1.6em;"> <tbody><tr> <td><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_D21.png?9bfb9" height="11" title="D21" alt="D21"></td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> (er, "[one] among" or possibly re, mouth) above a number to represent the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example: <table border="0" cellpadding="0.5em" style="margin-left: 1.6em;"> <tbody><tr> <td><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_D21.png?9bfb9" height="11" title="D21" alt="D21"> <img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_Z1.png?4dc06" height="16" title="Z1" alt="Z1"> <img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_Z1.png?4dc06" height="16" title="Z1" alt="Z1"> <img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_Z1.png?4dc06" height="16" title="Z1" alt="Z1"></td> </tr></tbody></table> </td></tr></tbody></table> </td> <td style="padding-right:1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{3}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c2d9944b78df9812dd9aaee35f9c12e3fecfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.452ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{3}}}"></noscript><span class="lazy-image-placeholder" style="width: 4.452ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c2d9944b78df9812dd9aaee35f9c12e3fecfef" data-alt="{\displaystyle ={\frac {1}{3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="border-left:1px solid black; padding-left:1em;"><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_D21.png?9bfb9" height="11" title="D21" alt="D21"> <img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_V20.png?e78cb" height="13" title="V20" alt="V20"></td> </tr></tbody></table> </td></tr></tbody></table> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{10}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38f3b3e71c3e22cf40e089d4dcea8ba2c23755b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.614ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{10}}}"></noscript><span class="lazy-image-placeholder" style="width: 5.614ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38f3b3e71c3e22cf40e089d4dcea8ba2c23755b" data-alt="{\displaystyle ={\frac {1}{10}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr></tbody></table> The Egyptians had special symbols for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.658ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" data-alt="{\displaystyle {\tfrac {1}{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.658ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" data-alt="{\displaystyle {\tfrac {2}{3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{4}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{4}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.658ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec6051ef87eb0dafdaeaacd61f340052fcbf2bf" data-alt="{\displaystyle {\tfrac {3}{4}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  that were used to reduce the size of numbers greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.658ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" data-alt="{\displaystyle {\tfrac {1}{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written as a sum of distinct unit fractions according to the usual Egyptian fraction notation. <table cellpadding="1em" style="margin-left: 1.6em;"> <tbody><tr> <td><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_Aa13.png?4d8fd" height="8" title="Aa13" alt="Aa13"></td></tr></tbody></table> </td></tr></tbody></table> </td> <td style="padding-right:1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a07892f777aa1effcc97d19841cb073d2f2e998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.452ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 4.452ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a07892f777aa1effcc97d19841cb073d2f2e998" data-alt="{\displaystyle ={\frac {1}{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="border-left:1px solid black; padding-left:1em;"><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_D22.png?0b8f1" height="18" title="D22" alt="D22"></td></tr></tbody></table> </td></tr></tbody></table> </td> <td style="padding-right:1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {2}{3}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29acd89ff50e9de30d8cbaf2a4a09b70b5dfba21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.452ex; height:5.176ex;" alt="{\displaystyle ={\frac {2}{3}}}"></noscript><span class="lazy-image-placeholder" style="width: 4.452ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29acd89ff50e9de30d8cbaf2a4a09b70b5dfba21" data-alt="{\displaystyle ={\frac {2}{3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="border-left:1px solid black; padding-left:1em;"><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_D23.png?f63be" height="30" title="D23" alt="D23"></td></tr></tbody></table> </td></tr></tbody></table> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {3}{4}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c05b2925600d290789cdbb5a8b61f39583c0d41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.452ex; height:5.176ex;" alt="{\displaystyle ={\frac {3}{4}}}"></noscript><span class="lazy-image-placeholder" style="width: 4.452ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c05b2925600d290789cdbb5a8b61f39583c0d41a" data-alt="{\displaystyle ={\frac {3}{4}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr></tbody></table> The Egyptians also used an alternative notation modified from the Old Kingdom to denote a special set of fractions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2^{k}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0534932a3a6b23dd2fc59ffa9a05b5f27181dd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:3.176ex;" alt="{\displaystyle 1/2^{k}}"></noscript><span class="lazy-image-placeholder" style="width: 4.576ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0534932a3a6b23dd2fc59ffa9a05b5f27181dd3" data-alt="{\displaystyle 1/2^{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,2,\dots ,6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,2,\dots ,6}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c95d4838e2a8e3ecd60e00fb516944609fcdc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.009ex; height:2.509ex;" alt="{\displaystyle k=1,2,\dots ,6}"></noscript><span class="lazy-image-placeholder" style="width: 14.009ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c95d4838e2a8e3ecd60e00fb516944609fcdc1" data-alt="{\displaystyle k=1,2,\dots ,6}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) and sums of these numbers, which are necessarily <a href="/wiki/Dyadic_rational" title="Dyadic rational">dyadic rational</a> numbers. These have been called "Horus-Eye fractions" after a theory (now discredited)<a href="#cite_note-4">[4]</a> that they were based on the parts of the <a href="/wiki/Eye_of_Horus" title="Eye of Horus">Eye of Horus</a> symbol. They were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a <a href="/wiki/Hekat_(volume_unit)" class="mw-redirect" title="Hekat (volume unit)">hekat</a>, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, as described in the <a href="/wiki/Akhmim_Wooden_Tablet" class="mw-redirect" title="Akhmim Wooden Tablet">Akhmim Wooden Tablet</a>. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the usual Egyptian fraction notation as multiples of a ro, a unit equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{320}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>320</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{320}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac2e7d800509f43af8b97e5bc0766c7792d2d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.302ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{320}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.302ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac2e7d800509f43af8b97e5bc0766c7792d2d77" data-alt="{\displaystyle {\tfrac {1}{320}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of a hekat. <div class="mw-heading mw-heading3"><h3 id="Calculation_methods">Calculation methods</h3> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=4" title="Edit section: Calculation methods" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{n}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada21e7887bd500f615e37467aa9a0780c48a983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.343ex;" alt="{\displaystyle {\tfrac {2}{n}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.822ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada21e7887bd500f615e37467aa9a0780c48a983" data-alt="{\displaystyle {\tfrac {2}{n}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. Additionally, the expansions in the table do not match any single identity; rather, different identities match the expansions for <a href="/wiki/Prime_number" title="Prime number">prime</a> and for <a href="/wiki/Composite_number" title="Composite number">composite</a> denominators, and more than one identity fits the numbers of each type: <ul><li>For small odd prime denominators <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.259ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" data-alt="{\displaystyle p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the expansion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{p}}={\frac {1}{(p+1)/2}}+{\frac {1}{p(p+1)/2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{p}}={\frac {1}{(p+1)/2}}+{\frac {1}{p(p+1)/2}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb009d36bae4c2fadd45ca74570c41ecd341dd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.399ex; height:6.009ex;" alt="{\displaystyle {\frac {2}{p}}={\frac {1}{(p+1)/2}}+{\frac {1}{p(p+1)/2}}}"></noscript><span class="lazy-image-placeholder" style="width: 29.399ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb009d36bae4c2fadd45ca74570c41ecd341dd5" data-alt="{\displaystyle {\frac {2}{p}}={\frac {1}{(p+1)/2}}+{\frac {1}{p(p+1)/2}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  was used.</li> <li>For larger prime denominators, an expansion of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{p}}={\frac {1}{A}}+{\frac {2A-p}{Ap}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>A</mi> <mo>−</mo> <mi>p</mi> </mrow> <mrow> <mi>A</mi> <mi>p</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{p}}={\frac {1}{A}}+{\frac {2A-p}{Ap}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec31b11ca85385c4ac680afdcd06127144b9c723" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.275ex; height:5.843ex;" alt="{\displaystyle {\frac {2}{p}}={\frac {1}{A}}+{\frac {2A-p}{Ap}}}"></noscript><span class="lazy-image-placeholder" style="width: 18.275ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec31b11ca85385c4ac680afdcd06127144b9c723" data-alt="{\displaystyle {\frac {2}{p}}={\frac {1}{A}}+{\frac {2A-p}{Ap}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  was used, where <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a number with many divisors (such as a <a href="/wiki/Practical_number" title="Practical number">practical number</a>) between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {p}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {p}{2}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48267343357b6bc51dd03744fcf8fa48679ce0d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.663ex; height:3.509ex;" alt="{\displaystyle {\tfrac {p}{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.663ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48267343357b6bc51dd03744fcf8fa48679ce0d4" data-alt="{\displaystyle {\tfrac {p}{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.259ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" data-alt="{\displaystyle p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The remaining term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2A-p)/Ap}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>A</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2A-p)/Ap}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32c4a35c0e3fae23f2b3bd11369b1deae5019a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.8ex; height:2.843ex;" alt="{\displaystyle (2A-p)/Ap}"></noscript><span class="lazy-image-placeholder" style="width: 12.8ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32c4a35c0e3fae23f2b3bd11369b1deae5019a0" data-alt="{\displaystyle (2A-p)/Ap}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  was expanded by representing the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2A-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>A</mi> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2A-p}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d08e4bc35eae269c60b00e019cf3a68872cbd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.915ex; height:2.509ex;" alt="{\displaystyle 2A-p}"></noscript><span class="lazy-image-placeholder" style="width: 6.915ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d08e4bc35eae269c60b00e019cf3a68872cbd0" data-alt="{\displaystyle 2A-p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as a sum of divisors 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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and forming a fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d}{Ap}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>d</mi> <mrow> <mi>A</mi> <mi>p</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {d}{Ap}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4bcb5eebcaac3dbcc08095766ea793dd21c868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:2.896ex; height:4.176ex;" alt="{\displaystyle {\tfrac {d}{Ap}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.896ex;height: 4.176ex;vertical-align: -1.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4bcb5eebcaac3dbcc08095766ea793dd21c868" data-alt="{\displaystyle {\tfrac {d}{Ap}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for each such divisor <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.216ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" data-alt="{\displaystyle d}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in this sum.<a href="#cite_note-5">[5]</a> As an example, Ahmes' expansion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{37}}={\tfrac {1}{24}}+{\tfrac {1}{111}}+{\tfrac {1}{296}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>37</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>111</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>296</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{37}}={\tfrac {1}{24}}+{\tfrac {1}{111}}+{\tfrac {1}{296}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e2315aa57b507ea5bc68e3cdd1a3620349b56f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.344ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{37}}={\tfrac {1}{24}}+{\tfrac {1}{111}}+{\tfrac {1}{296}}}"></noscript><span class="lazy-image-placeholder" style="width: 20.344ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e2315aa57b507ea5bc68e3cdd1a3620349b56f" data-alt="{\displaystyle {\tfrac {2}{37}}={\tfrac {1}{24}}+{\tfrac {1}{111}}+{\tfrac {1}{296}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  fits this pattern with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=24}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209067cdabc7888fca6790cfa9209e409cc0f40f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.166ex; height:2.176ex;" alt="{\displaystyle A=24}"></noscript><span class="lazy-image-placeholder" style="width: 7.166ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209067cdabc7888fca6790cfa9209e409cc0f40f" data-alt="{\displaystyle A=24}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2A-p=11=8+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>A</mi> <mo>−</mo> <mi>p</mi> <mo>=</mo> <mn>11</mn> <mo>=</mo> <mn>8</mn> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2A-p=11=8+3}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57de9e31b73d4e9ef00bca8b6225f994c7fd855d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.603ex; height:2.509ex;" alt="{\displaystyle 2A-p=11=8+3}"></noscript><span class="lazy-image-placeholder" style="width: 20.603ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57de9e31b73d4e9ef00bca8b6225f994c7fd855d" data-alt="{\displaystyle 2A-p=11=8+3}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{111}}={\tfrac {8}{24\cdot 37}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>111</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>8</mn> <mrow> <mn>24</mn> <mo>⋅</mo> <mn>37</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{111}}={\tfrac {8}{24\cdot 37}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c25fd7644673724f65e2088ecd011d33c7a450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.982ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{111}}={\tfrac {8}{24\cdot 37}}}"></noscript><span class="lazy-image-placeholder" style="width: 10.982ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c25fd7644673724f65e2088ecd011d33c7a450" data-alt="{\displaystyle {\tfrac {1}{111}}={\tfrac {8}{24\cdot 37}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{296}}={\tfrac {3}{24\cdot 37}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>296</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mrow> <mn>24</mn> <mo>⋅</mo> <mn>37</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{296}}={\tfrac {3}{24\cdot 37}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1efd574aef70c9e12af9f61711da97d5c33b2a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.982ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{296}}={\tfrac {3}{24\cdot 37}}}"></noscript><span class="lazy-image-placeholder" style="width: 10.982ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1efd574aef70c9e12af9f61711da97d5c33b2a12" data-alt="{\displaystyle {\tfrac {1}{296}}={\tfrac {3}{24\cdot 37}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . There may be many different expansions of this type for a given <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.259ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" data-alt="{\displaystyle p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ; however, as K. S. Brown observed, the expansion chosen by the Egyptians was often the one that caused the largest denominator to be as small as possible, among all expansions fitting this pattern.</li> <li>For some composite denominators, factored as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cdot 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\cdot q}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ccd43c59111042c75f1c0e10f31293fce7eb5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.007ex; height:2.009ex;" alt="{\displaystyle p\cdot q}"></noscript><span class="lazy-image-placeholder" style="width: 4.007ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ccd43c59111042c75f1c0e10f31293fce7eb5b8" data-alt="{\displaystyle p\cdot q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the expansion for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{pq}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{pq}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25089553761e835770674bb1826f668cd989e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.419ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{pq}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.419ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25089553761e835770674bb1826f668cd989e57" data-alt="{\displaystyle {\tfrac {2}{pq}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  has the form of an expansion for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{p}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5db639d0a9a55b6e59c6d61ff833f935a085fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.663ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{p}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.663ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5db639d0a9a55b6e59c6d61ff833f935a085fe" data-alt="{\displaystyle {\tfrac {2}{p}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with each denominator multiplied by <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.07ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" data-alt="{\displaystyle q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This method appears to have been used for many of the composite numbers in the Rhind papyrus,<a href="#cite_note-6">[6]</a> but there are exceptions, notably <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{35}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>35</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{35}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7772fe1cfcca8111f62832a2b2977f998a9d98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{35}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.48ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7772fe1cfcca8111f62832a2b2977f998a9d98" data-alt="{\displaystyle {\tfrac {2}{35}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{91}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>91</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{91}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7405ab9158bd9c14c99ef38636c4cf524bd09018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{91}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.48ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7405ab9158bd9c14c99ef38636c4cf524bd09018" data-alt="{\displaystyle {\tfrac {2}{91}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{95}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>95</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{95}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54323196655936cb195cb2f0f1d98d89ec9fece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{95}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.48ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54323196655936cb195cb2f0f1d98d89ec9fece" data-alt="{\displaystyle {\tfrac {2}{95}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .<a href="#cite_note-FOOTNOTEKnorr1982-7">[7]</a></li> <li>One can also expand <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{pq}}={\frac {1}{p(p+q)/2}}+{\frac {1}{q(p+q)/2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{pq}}={\frac {1}{p(p+q)/2}}+{\frac {1}{q(p+q)/2}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee9baa111dded5d5dd4ade4134efcb54f9f3211" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.999ex; height:6.009ex;" alt="{\displaystyle {\frac {2}{pq}}={\frac {1}{p(p+q)/2}}+{\frac {1}{q(p+q)/2}}.}"></noscript><span class="lazy-image-placeholder" style="width: 31.999ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee9baa111dded5d5dd4ade4134efcb54f9f3211" data-alt="{\displaystyle {\frac {2}{pq}}={\frac {1}{p(p+q)/2}}+{\frac {1}{q(p+q)/2}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For instance, Ahmes expands <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{35}}={\tfrac {2}{5\cdot 7}}={\tfrac {1}{30}}+{\tfrac {1}{42}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>35</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>7</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{35}}={\tfrac {2}{5\cdot 7}}={\tfrac {1}{30}}+{\tfrac {1}{42}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ef39c068d34e58ee6fe9c6d36bf4867276d77f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.415ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{35}}={\tfrac {2}{5\cdot 7}}={\tfrac {1}{30}}+{\tfrac {1}{42}}}"></noscript><span class="lazy-image-placeholder" style="width: 19.415ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ef39c068d34e58ee6fe9c6d36bf4867276d77f" data-alt="{\displaystyle {\tfrac {2}{35}}={\tfrac {2}{5\cdot 7}}={\tfrac {1}{30}}+{\tfrac {1}{42}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Later scribes used a more general form of this expansion, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{pq}}={\frac {1}{p(p+q)/n}}+{\frac {1}{q(p+q)/n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{pq}}={\frac {1}{p(p+q)/n}}+{\frac {1}{q(p+q)/n}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c6b9cd38b0eec359763a0a3b6ce5545e9273d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.464ex; height:6.009ex;" alt="{\displaystyle {\frac {n}{pq}}={\frac {1}{p(p+q)/n}}+{\frac {1}{q(p+q)/n}},}"></noscript><span class="lazy-image-placeholder" style="width: 32.464ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c6b9cd38b0eec359763a0a3b6ce5545e9273d8" data-alt="{\displaystyle {\frac {n}{pq}}={\frac {1}{p(p+q)/n}}+{\frac {1}{q(p+q)/n}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which works when <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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02fcb4cecca0b7e3116a7351e4345b48ef6de371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.169ex; height:2.343ex;" alt="{\displaystyle p+q}"></noscript><span class="lazy-image-placeholder" style="width: 5.169ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02fcb4cecca0b7e3116a7351e4345b48ef6de371" data-alt="{\displaystyle p+q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a multiple 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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .<a href="#cite_note-FOOTNOTEEves1953-8">[8]</a></li> <li>The final (prime) expansion in the Rhind papyrus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{101}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>101</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{101}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d73ac74059389a67c0cc3f1483e367e849dd93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.302ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{101}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.302ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d73ac74059389a67c0cc3f1483e367e849dd93" data-alt="{\displaystyle {\tfrac {2}{101}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , does not fit any of these forms, but instead uses an expansion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{p}}={\frac {1}{p}}+{\frac {1}{2p}}+{\frac {1}{3p}}+{\frac {1}{6p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mi>p</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{p}}={\frac {1}{p}}+{\frac {1}{2p}}+{\frac {1}{3p}}+{\frac {1}{6p}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4cc0d1d88e867e93c03be3b4000e9b95a344df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.135ex; height:5.676ex;" alt="{\displaystyle {\frac {2}{p}}={\frac {1}{p}}+{\frac {1}{2p}}+{\frac {1}{3p}}+{\frac {1}{6p}}}"></noscript><span class="lazy-image-placeholder" style="width: 25.135ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4cc0d1d88e867e93c03be3b4000e9b95a344df" data-alt="{\displaystyle {\frac {2}{p}}={\frac {1}{p}}+{\frac {1}{2p}}+{\frac {1}{3p}}+{\frac {1}{6p}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  that may be applied regardless of the value of <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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.259ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" data-alt="{\displaystyle p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{101}}={\tfrac {1}{101}}+{\tfrac {1}{202}}+{\tfrac {1}{303}}+{\tfrac {1}{606}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>101</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>101</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>202</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>303</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>606</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{101}}={\tfrac {1}{101}}+{\tfrac {1}{202}}+{\tfrac {1}{303}}+{\tfrac {1}{606}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4123b121da543b80d9d730dbb940e7e4a4982e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.13ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{101}}={\tfrac {1}{101}}+{\tfrac {1}{202}}+{\tfrac {1}{303}}+{\tfrac {1}{606}}}"></noscript><span class="lazy-image-placeholder" style="width: 28.13ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4123b121da543b80d9d730dbb940e7e4a4982e1f" data-alt="{\displaystyle {\tfrac {2}{101}}={\tfrac {1}{101}}+{\tfrac {1}{202}}+{\tfrac {1}{303}}+{\tfrac {1}{606}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . A related expansion was also used in the Egyptian Mathematical Leather Roll for several cases.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Later_usage">Later usage</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=5" title="Edit section: Later usage" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a> and <a href="/wiki/Greedy_algorithm_for_Egyptian_fractions" title="Greedy algorithm for Egyptian fractions">Greedy algorithm for Egyptian fractions</a></div> Egyptian fraction notation continued to be used in Greek times and into the Middle Ages,<a href="#cite_note-FOOTNOTEStruik1967-9">[9]</a> despite complaints as early as <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>'s <a href="/wiki/Almagest" title="Almagest">Almagest</a> about the clumsiness of the notation compared to alternatives such as the <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian</a> <a href="/wiki/Sexagesimal" title="Sexagesimal">base-60 notation</a>. Related problems of decomposition into unit fractions were also studied in 9th-century India by Jain mathematician <a href="/wiki/Mah%C4%81v%C4%ABra_(mathematician)" title="Mahāvīra (mathematician)">Mahāvīra</a>.<a href="#cite_note-FOOTNOTEKusuba2004-10">[10]</a> An important text of medieval European mathematics, the <a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a> (1202) of <a href="/wiki/Leonardo_of_Pisa" class="mw-redirect" title="Leonardo of Pisa">Leonardo of Pisa</a> (more commonly known as Fibonacci), provides some insight into the uses of Egyptian fractions in the Middle Ages, and introduces topics that continue to be important in modern mathematical study of these series. The primary subject of the Liber Abaci is calculations involving decimal and vulgar fraction notation, which eventually replaced Egyptian fractions. Fibonacci himself used a complex notation for fractions involving a combination of a <a href="/wiki/Mixed_radix" title="Mixed radix">mixed radix</a> notation with sums of fractions. Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book<a href="#cite_note-11">[11]</a> provides a list of methods for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum of divisors of the denominator; this is possible whenever the denominator is a <a href="/wiki/Practical_number" title="Practical number">practical number</a>, and Liber Abaci includes tables of expansions of this type for the practical numbers 6, 8, 12, 20, 24, 60, and 100. The next several methods involve algebraic identities such as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c2726aca6139e0a9ea3836ae25f737bba3c43e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.524ex; height:6.009ex;" alt="{\displaystyle {\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}.}"></noscript><span class="lazy-image-placeholder" style="width: 25.524ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c2726aca6139e0a9ea3836ae25f737bba3c43e" data-alt="{\displaystyle {\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For instance, Fibonacci represents the fraction <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠8/11⁠ by splitting the numerator into a sum of two numbers, each of which divides one plus the denominator: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠8/11⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/11⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/11⁠. Fibonacci applies the algebraic identity above to each these two parts, producing the expansion <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠8/11⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/22⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/66⁠. Fibonacci describes similar methods for denominators that are two or three less than a number with many factors. In the rare case that these other methods all fail, Fibonacci suggests a <a href="/wiki/Greedy_algorithm_for_Egyptian_fractions" title="Greedy algorithm for Egyptian fractions">"greedy" algorithm</a> for computing Egyptian fractions, in which one repeatedly chooses the unit fraction with the smallest denominator that is no larger than the remaining fraction to be expanded: that is, in more modern notation, we replace a fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/y⁠ by the expansion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{y}}={\frac {1}{\,\left\lceil {\frac {y}{x}}\right\rceil \,}}+{\frac {(-y)\,{\bmod {\,}}x}{y\left\lceil {\frac {y}{x}}\right\rceil }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mspace width="thinmathspace"></mspace> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace"></mspace> </mrow> </mrow> <mi>x</mi> </mrow> <mrow> <mi>y</mi> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{y}}={\frac {1}{\,\left\lceil {\frac {y}{x}}\right\rceil \,}}+{\frac {(-y)\,{\bmod {\,}}x}{y\left\lceil {\frac {y}{x}}\right\rceil }},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87982e5702e256819b677467e5921d1009f9415" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.727ex; height:7.009ex;" alt="{\displaystyle {\frac {x}{y}}={\frac {1}{\,\left\lceil {\frac {y}{x}}\right\rceil \,}}+{\frac {(-y)\,{\bmod {\,}}x}{y\left\lceil {\frac {y}{x}}\right\rceil }},}"></noscript><span class="lazy-image-placeholder" style="width: 27.727ex;height: 7.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87982e5702e256819b677467e5921d1009f9415" data-alt="{\displaystyle {\frac {x}{y}}={\frac {1}{\,\left\lceil {\frac {y}{x}}\right\rceil \,}}+{\frac {(-y)\,{\bmod {\,}}x}{y\left\lceil {\frac {y}{x}}\right\rceil }},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where ⌈ ⌉ represents the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">ceiling function</a>; since (−y) mod x < x, this method yields a finite expansion. Fibonacci suggests switching to another method after the first such expansion, but he also gives examples in which this greedy expansion was iterated until a complete Egyptian fraction expansion was constructed: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/13⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/18⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/468⁠ and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠17/29⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/12⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/348⁠. Compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators, and Fibonacci himself noted the awkwardness of the expansions produced by this method. For instance, the greedy method expands <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>121</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>25</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>757</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>763</mn> <mspace width="thinmathspace"></mspace> <mn>309</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>873</mn> <mspace width="thinmathspace"></mspace> <mn>960</mn> <mspace width="thinmathspace"></mspace> <mn>180</mn> <mspace width="thinmathspace"></mspace> <mn>913</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mn>527</mn> <mspace width="thinmathspace"></mspace> <mn>612</mn> <mspace width="thinmathspace"></mspace> <mn>795</mn> <mspace width="thinmathspace"></mspace> <mn>642</mn> <mspace width="thinmathspace"></mspace> <mn>093</mn> <mspace width="thinmathspace"></mspace> <mn>418</mn> <mspace width="thinmathspace"></mspace> <mn>846</mn> <mspace width="thinmathspace"></mspace> <mn>225</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5af6b710b559d471480c4ddce74c984a7d5f6561" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:84.054ex; height:5.343ex;" alt="{\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},}"></noscript><span class="lazy-image-placeholder" style="width: 84.054ex;height: 5.343ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5af6b710b559d471480c4ddce74c984a7d5f6561" data-alt="{\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  while other methods lead to the shorter expansion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>121</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>33</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>121</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>363</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5d67bd2e7732b3f29aeb5e545d36c02cec03cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.558ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.}"></noscript><span class="lazy-image-placeholder" style="width: 25.558ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5d67bd2e7732b3f29aeb5e545d36c02cec03cd" data-alt="{\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <a href="/wiki/Sylvester%27s_sequence" title="Sylvester's sequence">Sylvester's sequence</a> 2, 3, 7, 43, 1807, ... can be viewed as generated by an infinite greedy expansion of this type for the number 1, where at each step we choose the denominator ⌊ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠y/x⁠ ⌋ + 1 instead of ⌈ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠y/x⁠ ⌉, and sometimes Fibonacci's greedy algorithm is attributed to <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">James Joseph Sylvester</a>. After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠a/b⁠ by searching for a number c having many divisors, with <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠b/2⁠ < c < b, replacing <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠a/b⁠ by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ac/bc⁠, and expanding ac as a sum of divisors of bc, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Modern_number_theory">Modern number theory</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=6" title="Edit section: Modern number theory" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Erd%C5%91s%E2%80%93Graham_problem" title="Erdős–Graham problem">Erdős–Graham problem</a>, <a href="/wiki/Zn%C3%A1m%27s_problem" title="Znám's problem">Znám's problem</a>, and <a href="/wiki/Engel_expansion" title="Engel expansion">Engel expansion</a></div> Although Egyptian fractions are no longer used in most practical applications of mathematics, modern number theorists have continued to study many different problems related to them. These include problems of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of various methods for Egyptian fraction expansion, and showing that expansions exist for any sufficiently dense set of sufficiently <a href="/wiki/Smooth_number" title="Smooth number">smooth numbers</a>. <ul><li>One of the earliest publications of <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> proved that it is not possible for a <a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">harmonic progression</a> to form an Egyptian fraction representation of an <a href="/wiki/Integer" title="Integer">integer</a>. The reason is that, necessarily, at least one denominator of the progression will be divisible by a <a href="/wiki/Prime_number" title="Prime number">prime number</a> that does not divide any other denominator.<a href="#cite_note-12">[12]</a> The latest publication of Erdős, nearly 20 years after his death, proves that every integer has a representation in which all denominators are products of three primes.<a href="#cite_note-FOOTNOTEButlerErd%C5%91sGraham2015-13">[13]</a></li> <li>The <a href="/wiki/Erd%C5%91s%E2%80%93Graham_conjecture" class="mw-redirect" title="Erdős–Graham conjecture">Erdős–Graham conjecture</a> in <a href="/wiki/Number_theory" title="Number theory">combinatorial number theory</a> states that, if the integers greater than 1 are partitioned into finitely many subsets, then one of the subsets has a finite subset of itself whose reciprocals sum to one. That is, for every r > 0, and every r-coloring of the integers greater than one, there is a finite monochromatic subset S of these integers such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in S}{\frac {1}{n}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in S}{\frac {1}{n}}=1.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9f2a52b8f8c6f40f960509e48c3e9a6598b2fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.881ex; height:6.509ex;" alt="{\displaystyle \sum _{n\in S}{\frac {1}{n}}=1.}"></noscript><span class="lazy-image-placeholder" style="width: 10.881ex;height: 6.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9f2a52b8f8c6f40f960509e48c3e9a6598b2fa" data-alt="{\displaystyle \sum _{n\in S}{\frac {1}{n}}=1.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The conjecture was proven in 2003 by <a href="/wiki/Ernest_S._Croot_III" title="Ernest S. Croot III">Ernest S. Croot III</a>.</li> <li><a href="/wiki/Zn%C3%A1m%27s_problem" title="Znám's problem">Znám's problem</a> and <a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">primary pseudoperfect numbers</a> are closely related to the existence of Egyptian fractions of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53038678de69a451020554b89282983361975627" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.778ex; height:5.509ex;" alt="{\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.}"></noscript><span class="lazy-image-placeholder" style="width: 20.778ex;height: 5.509ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53038678de69a451020554b89282983361975627" data-alt="{\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For instance, the primary pseudoperfect number 1806 is the product of the prime numbers 2, 3, 7, and 43, and gives rise to the Egyptian fraction 1 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/7⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/43⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/1806⁠.</li> <li>Egyptian fractions are normally defined as requiring all denominators to be distinct, but this requirement can be relaxed to allow repeated denominators. However, this relaxed form of Egyptian fractions does not allow for any number to be represented using fewer fractions, as any expansion with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {2}{k+1}}+{\frac {2}{k(k+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {2}{k+1}}+{\frac {2}{k(k+1)}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d89051c04499c36d6c85f45064f5aa891757344" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.995ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {2}{k+1}}+{\frac {2}{k(k+1)}}}"></noscript><span class="lazy-image-placeholder" style="width: 27.995ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d89051c04499c36d6c85f45064f5aa891757344" data-alt="{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {2}{k+1}}+{\frac {2}{k(k+1)}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  if k is odd, or simply by replacing <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/k⁠ by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/k⁠ if k is even. This result was first proven by <a href="#CITEREFTakenouchi1921">Takenouchi (1921)</a>.</li> <li>Graham and Jewett<a href="#cite_note-14">[14]</a> proved that it is similarly possible to convert expansions with repeated denominators to (longer) Egyptian fractions, via the replacement <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {1}{k}}+{\frac {1}{k+1}}+{\frac {1}{k(k+1)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {1}{k}}+{\frac {1}{k+1}}+{\frac {1}{k(k+1)}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82627ffb047088f90131d1767ea4614c96dfed3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.53ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {1}{k}}+{\frac {1}{k+1}}+{\frac {1}{k(k+1)}}.}"></noscript><span class="lazy-image-placeholder" style="width: 33.53ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82627ffb047088f90131d1767ea4614c96dfed3" data-alt="{\displaystyle {\frac {1}{k}}+{\frac {1}{k}}={\frac {1}{k}}+{\frac {1}{k+1}}+{\frac {1}{k(k+1)}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This method can lead to long expansions with large denominators, such as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{5}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{31}}+{\frac {1}{32}}+{\frac {1}{42}}+{\frac {1}{43}}+{\frac {1}{56}}+{\frac {1}{930}}+{\frac {1}{931}}+{\frac {1}{992}}+{\frac {1}{1806}}+{\frac {1}{865\,830}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>31</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>32</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>43</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>56</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>930</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>931</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>992</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1806</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>865</mn> <mspace width="thinmathspace"></mspace> <mn>830</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{5}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{31}}+{\frac {1}{32}}+{\frac {1}{42}}+{\frac {1}{43}}+{\frac {1}{56}}+{\frac {1}{930}}+{\frac {1}{931}}+{\frac {1}{992}}+{\frac {1}{1806}}+{\frac {1}{865\,830}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b538d88c5c26b5876d284c2c531fb611b03d0810" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:99.124ex; height:5.343ex;" alt="{\displaystyle {\frac {4}{5}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{31}}+{\frac {1}{32}}+{\frac {1}{42}}+{\frac {1}{43}}+{\frac {1}{56}}+{\frac {1}{930}}+{\frac {1}{931}}+{\frac {1}{992}}+{\frac {1}{1806}}+{\frac {1}{865\,830}}.}"></noscript><span class="lazy-image-placeholder" style="width: 99.124ex;height: 5.343ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b538d88c5c26b5876d284c2c531fb611b03d0810" data-alt="{\displaystyle {\frac {4}{5}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{31}}+{\frac {1}{32}}+{\frac {1}{42}}+{\frac {1}{43}}+{\frac {1}{56}}+{\frac {1}{930}}+{\frac {1}{931}}+{\frac {1}{992}}+{\frac {1}{1806}}+{\frac {1}{865\,830}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <a href="#CITEREFBotts1967">Botts (1967)</a> had originally used this replacement technique to show that any rational number has Egyptian fraction representations with arbitrarily large minimum denominators.</li> <li>Any fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/y⁠ has an Egyptian fraction representation in which the maximum denominator is bounded by<a href="#cite_note-FOOTNOTEYokota1988-15">[15]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left(y\log y\left(\log \log y\right)^{4}\left(\log \log \log y\right)^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mi>log</mi> <mo>⁡</mo> <mi>y</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left(y\log y\left(\log \log y\right)^{4}\left(\log \log \log y\right)^{2}\right),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da50926eafb6c20429130322941c220457b2fb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.859ex; height:4.843ex;" alt="{\displaystyle O\left(y\log y\left(\log \log y\right)^{4}\left(\log \log \log y\right)^{2}\right),}"></noscript><span class="lazy-image-placeholder" style="width: 36.859ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da50926eafb6c20429130322941c220457b2fb1" data-alt="{\displaystyle O\left(y\log y\left(\log \log y\right)^{4}\left(\log \log \log y\right)^{2}\right),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and a representation with at most <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\left({\sqrt {\log y}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>log</mi> <mo>⁡</mo> <mi>y</mi> </msqrt> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O\left({\sqrt {\log y}}\right)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9e637e327cef89bab1e05afb8a161e4d1b928b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.128ex; height:3.509ex;" alt="{\displaystyle O\left({\sqrt {\log y}}\right)}"></noscript><span class="lazy-image-placeholder" style="width: 11.128ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9e637e327cef89bab1e05afb8a161e4d1b928b" data-alt="{\displaystyle O\left({\sqrt {\log y}}\right)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  terms.<a href="#cite_note-FOOTNOTEVose1985-16">[16]</a> The number of terms must sometimes be at least proportional to log log y; for instance this is true for the fractions in the sequence <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/3⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/7⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠42/43⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1806/1807⁠, ... whose denominators form <a href="/wiki/Sylvester%27s_sequence" title="Sylvester's sequence">Sylvester's sequence</a>. It has been conjectured that O(log log y) terms are always enough.<a href="#cite_note-FOOTNOTEErd%C5%91s1950-17">[17]</a> It is also possible to find representations in which both the maximum denominator and the number of terms are small.<a href="#cite_note-FOOTNOTETenenbaumYokota1990-18">[18]</a></li> <li><a href="#CITEREFGraham1964">Graham (1964)</a> characterized the numbers that can be represented by Egyptian fractions in which all denominators are nth powers. In particular, a rational number q can be represented as an Egyptian fraction with square denominators if and only if q lies in one of the two half-open intervals <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[0,{\frac {\pi ^{2}}{6}}-1\right)\cup \left[1,{\frac {\pi ^{2}}{6}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>∪</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[0,{\frac {\pi ^{2}}{6}}-1\right)\cup \left[1,{\frac {\pi ^{2}}{6}}\right).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3246ea878a54c8288d3bd2b60ce87873d0e31c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.337ex; height:6.343ex;" alt="{\displaystyle \left[0,{\frac {\pi ^{2}}{6}}-1\right)\cup \left[1,{\frac {\pi ^{2}}{6}}\right).}"></noscript><span class="lazy-image-placeholder" style="width: 24.337ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3246ea878a54c8288d3bd2b60ce87873d0e31c" data-alt="{\displaystyle \left[0,{\frac {\pi ^{2}}{6}}-1\right)\cup \left[1,{\frac {\pi ^{2}}{6}}\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </li> <li><a href="#CITEREFMartin1999">Martin (1999)</a> showed that any rational number has very dense expansions, using a constant fraction of the denominators up to N for any sufficiently large N.</li> <li><a href="/wiki/Engel_expansion" title="Engel expansion">Engel expansion</a>, sometimes called an Egyptian product, is a form of Egyptian fraction expansion in which each denominator is a multiple of the previous one: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/484bc63e1bf72f9cc5adcbfcedc20887364be6d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.919ex; height:5.509ex;" alt="{\displaystyle x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .}"></noscript><span class="lazy-image-placeholder" style="width: 32.919ex;height: 5.509ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/484bc63e1bf72f9cc5adcbfcedc20887364be6d0" data-alt="{\displaystyle x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  In addition, the sequence of multipliers ai is required to be nondecreasing. Every rational number has a finite Engel expansion, while <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> have an infinite Engel expansion.</li> <li><a href="#CITEREFAnshelGoldfeld1991">Anshel & Goldfeld (1991)</a> study numbers that have multiple distinct Egyptian fraction representations with the same number of terms and the same product of denominators; for instance, one of the examples they supply is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5}{12}}={\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{15}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{20}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>20</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5}{12}}={\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{15}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{20}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b4d43cf7386b40f280646067b04cc932355984" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.845ex; height:5.176ex;" alt="{\displaystyle {\frac {5}{12}}={\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{15}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{20}}.}"></noscript><span class="lazy-image-placeholder" style="width: 36.845ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b4d43cf7386b40f280646067b04cc932355984" data-alt="{\displaystyle {\frac {5}{12}}={\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{15}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{20}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Unlike the ancient Egyptians, they allow denominators to be repeated in these expansions. They apply their results for this problem to the characterization of <a href="/wiki/Free_product" title="Free product">free products</a> of <a href="/wiki/Abelian_group" title="Abelian group">Abelian groups</a> by a small number of numerical parameters: the rank of the <a href="/wiki/Commutator_subgroup" title="Commutator subgroup">commutator subgroup</a>, the number of terms in the free product, and the product of the orders of the factors.</li> <li>The number of different n-term Egyptian fraction representations of the number one is bounded above and below by <a href="/wiki/Double_exponential_function" title="Double exponential function">double exponential functions</a> of n.<a href="#cite_note-FOOTNOTEKonyagin2014-19">[19]</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Open_problems">Open problems</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=7" title="Edit section: Open problems" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Odd_greedy_expansion" title="Odd greedy expansion">Odd greedy expansion</a> and <a href="/wiki/Erd%C5%91s%E2%80%93Straus_conjecture" title="Erdős–Straus conjecture">Erdős–Straus conjecture</a></div> Some notable problems remain unsolved with regard to Egyptian fractions, despite considerable effort by mathematicians. <ul><li>The <a href="/wiki/Erd%C5%91s%E2%80%93Straus_conjecture" title="Erdős–Straus conjecture">Erdős–Straus conjecture</a><a href="#cite_note-FOOTNOTEErd%C5%91s1950-17">[17]</a> concerns the length of the shortest expansion for a fraction of the form <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/n⁠. Does an expansion <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18712cca1c0fe53694bd863d0b1d5ea9aa8f07dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.173ex; height:5.676ex;" alt="{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.173ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18712cca1c0fe53694bd863d0b1d5ea9aa8f07dc" data-alt="{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  exist for every n? It is known to be true for all n < 1017, and for all but a vanishingly small fraction of possible values of n, but the general truth of the conjecture remains unknown.</li> <li>It is unknown whether an <a href="/wiki/Odd_greedy_expansion" title="Odd greedy expansion">odd greedy expansion</a> exists for every fraction with an odd denominator. If Fibonacci's greedy method is modified so that it always chooses the smallest possible odd denominator, under what conditions does this modified algorithm produce a finite expansion? An obvious necessary condition is that the starting fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/y⁠ have an odd denominator y, and it is conjectured but not known that this is also a sufficient condition. It is known<a href="#cite_note-20">[20]</a> that every <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/y⁠ with odd y has an expansion into distinct odd unit fractions, constructed using a different method than the greedy algorithm.</li> <li>It is possible to use <a href="/wiki/Brute-force_search" title="Brute-force search">brute-force search</a> algorithms to find the Egyptian fraction representation of a given number with the fewest possible terms<a href="#cite_note-FOOTNOTEStewart1992-21">[21]</a> or minimizing the largest denominator; however, such algorithms can be quite inefficient. The existence of <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> algorithms for these problems, or more generally the <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">computational complexity</a> of such problems, remains unknown.</li></ul> <a href="#CITEREFGuy2004">Guy (2004)</a> describes these problems in more detail and lists numerous additional open problems. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=8" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><a href="/wiki/List_of_sums_of_reciprocals" title="List of sums of reciprocals">List of sums of reciprocals</a></li> <li><a href="/wiki/17-animal_inheritance_puzzle" title="17-animal inheritance puzzle">17-animal inheritance puzzle</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Notes">Notes</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=9" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <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-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFDickOgle2018">Dick & Ogle (2018)</a>; <a href="#CITEREFKoshalevaKreinovich2021">Koshaleva & Kreinovich (2021)</a> </li> <li id="cite_note-FOOTNOTEWilsonEdgingtonNguyenPescosolido2011-2"><a href="#cite_ref-FOOTNOTEWilsonEdgingtonNguyenPescosolido2011_2-0">^</a> <a href="#CITEREFWilsonEdgingtonNguyenPescosolido2011">Wilson et al. (2011)</a>. </li> <li id="cite_note-FOOTNOTEWinkler2004-3"><a href="#cite_ref-FOOTNOTEWinkler2004_3-0">^</a> <a href="#CITEREFWinkler2004">Winkler (2004)</a>. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFRitter2002">Ritter (2002)</a>. See also <a href="#CITEREFKatz2007">Katz (2007)</a> and <a href="#CITEREFRobsonStedall2009">Robson & Stedall (2009)</a>. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFHultsch1895">Hultsch (1895)</a>; <a href="#CITEREFBruins1957">Bruins (1957)</a> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFGillings1982">Gillings (1982)</a>; <a href="#CITEREFGardner2002">Gardner (2002)</a> </li> <li id="cite_note-FOOTNOTEKnorr1982-7"><a href="#cite_ref-FOOTNOTEKnorr1982_7-0">^</a> <a href="#CITEREFKnorr1982">Knorr (1982)</a>. </li> <li id="cite_note-FOOTNOTEEves1953-8"><a href="#cite_ref-FOOTNOTEEves1953_8-0">^</a> <a href="#CITEREFEves1953">Eves (1953)</a>. </li> <li id="cite_note-FOOTNOTEStruik1967-9"><a href="#cite_ref-FOOTNOTEStruik1967_9-0">^</a> <a href="#CITEREFStruik1967">Struik (1967)</a>. </li> <li id="cite_note-FOOTNOTEKusuba2004-10"><a href="#cite_ref-FOOTNOTEKusuba2004_10-0">^</a> <a href="#CITEREFKusuba2004">Kusuba (2004)</a>. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFSigler2002">Sigler (2002)</a>, chapter II.7 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFErd%C5%91s1932">Erdős (1932)</a>; <a href="#CITEREFGraham2013">Graham (2013)</a> </li> <li id="cite_note-FOOTNOTEButlerErdősGraham2015-13"><a href="#cite_ref-FOOTNOTEButlerErd%C5%91sGraham2015_13-0">^</a> <a href="#CITEREFButlerErd%C5%91sGraham2015">Butler, Erdős & Graham (2015)</a>. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> See <a href="#CITEREFWagon1999">Wagon (1999)</a> and <a href="#CITEREFBeeckmans1993">Beeckmans (1993)</a> </li> <li id="cite_note-FOOTNOTEYokota1988-15"><a href="#cite_ref-FOOTNOTEYokota1988_15-0">^</a> <a href="#CITEREFYokota1988">Yokota (1988)</a>. </li> <li id="cite_note-FOOTNOTEVose1985-16"><a href="#cite_ref-FOOTNOTEVose1985_16-0">^</a> <a href="#CITEREFVose1985">Vose (1985)</a>. </li> <li id="cite_note-FOOTNOTEErdős1950-17">^ <a href="#cite_ref-FOOTNOTEErd%C5%91s1950_17-0">a</a> <a href="#cite_ref-FOOTNOTEErd%C5%91s1950_17-1">b</a> <a href="#CITEREFErd%C5%91s1950">Erdős (1950)</a>. </li> <li id="cite_note-FOOTNOTETenenbaumYokota1990-18"><a href="#cite_ref-FOOTNOTETenenbaumYokota1990_18-0">^</a> <a href="#CITEREFTenenbaumYokota1990">Tenenbaum & Yokota (1990)</a>. </li> <li id="cite_note-FOOTNOTEKonyagin2014-19"><a href="#cite_ref-FOOTNOTEKonyagin2014_19-0">^</a> <a href="#CITEREFKonyagin2014">Konyagin (2014)</a>. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFBreusch1954">Breusch (1954)</a>; <a href="#CITEREFStewart1954">Stewart (1954)</a> </li> <li id="cite_note-FOOTNOTEStewart1992-21"><a href="#cite_ref-FOOTNOTEStewart1992_21-0">^</a> <a href="#CITEREFStewart1992">Stewart (1992)</a>. </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=10" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><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 .cs1-maint{color:#18911f}}</style><cite id="CITEREFAnshelGoldfeld1991" class="citation cs2">Anshel, Michael M.; <a href="/wiki/Dorian_M._Goldfeld" title="Dorian M. 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width:24.006ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}={\frac {a}{b}}}"></noscript><span class="lazy-image-placeholder" style="width: 24.006ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99e502109cf8ffe632119234daae5da79be54b6" data-alt="{\displaystyle \textstyle {\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}={\frac {a}{b}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  egyenlet egész számú megoldásairól"</a> [On a Diophantine equation] (PDF), Matematikai Lapok (in Hungarian), 1: 192–210, <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=0043117">0043117</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Matematikai+Lapok&rft.atitle=Az+MATH+RENDER+ERROR+egyenlet+eg%C3%A9sz+sz%C3%A1m%C3%BA+megold%C3%A1sair%C3%B3l&rft.volume=1&rft.pages=192-210&rft.date=1950&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0043117%23id-name%3DMR&rft.aulast=Erd%C5%91s&rft.aufirst=P%C3%A1l&rft_id=https%3A%2F%2Fwww.renyi.hu%2F~p_erdos%2F1950-02.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1953" class="citation cs2"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard</a> (1953), An Introduction to the History of Mathematics, Holt, Reinhard, and Winston, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-029558-0" title="Special:BookSources/0-03-029558-0"><bdi>0-03-029558-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+History+of+Mathematics&rft.pub=Holt%2C+Reinhard%2C+and+Winston&rft.date=1953&rft.isbn=0-03-029558-0&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner2002" class="citation cs2">Gardner, Milo (2002), "The Egyptian Mathematical Leather Roll, attested short term and long term", in <a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Gratton-Guinness, Ivor</a> (ed.), History of the Mathematical Sciences, Hindustan Book Co, pp. 119–134, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-85931-45-3" title="Special:BookSources/81-85931-45-3"><bdi>81-85931-45-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Egyptian+Mathematical+Leather+Roll%2C+attested+short+term+and+long+term&rft.btitle=History+of+the+Mathematical+Sciences&rft.pages=119-134&rft.pub=Hindustan+Book+Co&rft.date=2002&rft.isbn=81-85931-45-3&rft.aulast=Gardner&rft.aufirst=Milo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillings1982" class="citation cs2">Gillings, Richard J. 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Holt; Edgington, Cynthia P.; Nguyen, Kenny H.; Pescosolido, Ryan C.; Confrey, Jere (November 2011), "Fractions: how to fair share", Mathematics Teaching in the Middle School, 17 (4): 230–236, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fmathteacmiddscho.17.4.0230">10.5951/mathteacmiddscho.17.4.0230</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/10.5951/mathteacmiddscho.17.4.0230">10.5951/mathteacmiddscho.17.4.0230</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Teaching+in+the+Middle+School&rft.atitle=Fractions%3A+how+to+fair+share&rft.volume=17&rft.issue=4&rft.pages=230-236&rft.date=2011-11&rft_id=info%3Adoi%2F10.5951%2Fmathteacmiddscho.17.4.0230&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F10.5951%2Fmathteacmiddscho.17.4.0230%23id-name%3DJSTOR&rft.aulast=Wilson&rft.aufirst=P.+Holt&rft.au=Edgington%2C+Cynthia+P.&rft.au=Nguyen%2C+Kenny+H.&rft.au=Pescosolido%2C+Ryan+C.&rft.au=Confrey%2C+Jere&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWinkler2004" class="citation cs2"><a href="/wiki/Peter_Winkler" title="Peter Winkler">Winkler, Peter</a> (2004), "Uses of fuses", Mathematical Puzzles: A Connoisseur's Collection, A K Peters, pp. 2, 6, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-201-9" title="Special:BookSources/1-56881-201-9"><bdi>1-56881-201-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Uses+of+fuses&rft.btitle=Mathematical+Puzzles%3A+A+Connoisseur%27s+Collection&rft.pages=2%2C+6&rft.pub=A+K+Peters&rft.date=2004&rft.isbn=1-56881-201-9&rft.aulast=Winkler&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYokota1988" class="citation cs2">Yokota, Hisashi (1988), "On a problem of Bleicher and Erdős", Journal of Number Theory, 30 (2): 198–207, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-314X%2888%2990017-0">10.1016/0022-314X(88)90017-0</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=0961916">0961916</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+Number+Theory&rft.atitle=On+a+problem+of+Bleicher+and+Erd%C5%91s&rft.volume=30&rft.issue=2&rft.pages=198-207&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0022-314X%2888%2990017-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D961916%23id-name%3DMR&rft.aulast=Yokota&rft.aufirst=Hisashi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Egyptian_fraction&action=edit&section=11" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown" class="citation cs2">Brown, Kevin, <a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath340/kmath340.htm">Egyptian Unit Fractions</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Egyptian+Unit+Fractions&rft.aulast=Brown&rft.aufirst=Kevin&rft_id=http%3A%2F%2Fwww.mathpages.com%2Fhome%2Fkmath340%2Fkmath340.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEppstein" class="citation cs2"><a href="/wiki/David_Eppstein" title="David Eppstein">Eppstein, David</a>, <a rel="nofollow" class="external text" href="http://www.ics.uci.edu/~eppstein/numth/egypt/">Egyptian Fractions</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Egyptian+Fractions&rft.aulast=Eppstein&rft.aufirst=David&rft_id=http%3A%2F%2Fwww.ics.uci.edu%2F~eppstein%2Fnumth%2Fegypt%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnott" class="citation cs2">Knott, Ron, <a rel="nofollow" class="external text" href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html">Egyptian fractions</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Egyptian+fractions&rft.aulast=Knott&rft.aufirst=Ron&rft_id=http%3A%2F%2Fwww.maths.surrey.ac.uk%2Fhosted-sites%2FR.Knott%2FFractions%2Fegyptian.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/EgyptianFraction.html">"Egyptian Fraction"</a>, <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Egyptian+Fraction&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FEgyptianFraction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiroux" class="citation cs2">Giroux, André, <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/EgyptianFractions/">Egyptian Fractions</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Egyptian+Fractions&rft.aulast=Giroux&rft.aufirst=Andr%C3%A9&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FEgyptianFractions%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988"> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZeleny" class="citation cs2">Zeleny, Enrique, <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/AlgorithmsForEgyptianFractions/">Algorithms for Egyptian Fractions</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+for+Egyptian+Fractions&rft.aulast=Zeleny&rft.aufirst=Enrique&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FAlgorithmsForEgyptianFractions%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEgyptian+fraction" class="Z3988">, <a href="/wiki/The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a>, based on programs by <a href="/wiki/David_Eppstein" title="David Eppstein">David Eppstein</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div>    </section></div> <noscript><img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Egyptian_fraction&oldid=1253900926">https://en.wikipedia.org/w/index.php?title=Egyptian_fraction&oldid=1253900926</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Egyptian_fraction&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 28 October 2024, at 13:40 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%B3%D8%B1_%D9%85%D8%B5%D8%B1%D9%8A" title="كسر مصري – Arabic" lang="ar" hreflang="ar" data-title="كسر مصري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%B3%D0%B8%D0%BF%D0%B5%D1%82%D1%81%D0%BA%D0%B0_%D0%B4%D1%80%D0%BE%D0%B1" title="Египетска дроб – Bulgarian" lang="bg" hreflang="bg" data-title="Египетска дроб" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Fracci%C3%B3_eg%C3%ADpcia" title="Fracció egípcia – Catalan" lang="ca" hreflang="ca" data-title="Fracció egípcia" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fracci%C3%B3n_egipcia" title="Fracción egipcia – Spanish" lang="es" hreflang="es" data-title="Fracción egipcia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fraction_%C3%A9gyptienne" title="Fraction égyptienne – French" lang="fr" hreflang="fr" data-title="Fraction égyptienne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Fracci%C3%B3n_exipcia" title="Fracción exipcia – Galician" lang="gl" hreflang="gl" data-title="Fracción exipcia" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%A7%91%ED%8A%B8_%EB%B6%84%EC%88%98" title="이집트 분수 – Korean" lang="ko" hreflang="ko" data-title="이집트 분수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%D5%A3%D5%AB%D5%BA%D5%BF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AF%D5%B8%D5%BF%D5%B8%D6%80%D5%A1%D5%AF" title="Եգիպտական կոտորակ – Armenian" lang="hy" hreflang="hy" data-title="Եգիպտական կոտորակ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Frazione_egizia" title="Frazione egizia – Italian" lang="it" hreflang="it" data-title="Frazione egizia" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%91%D7%A8_%D7%9E%D7%A6%D7%A8%D7%99" title="שבר מצרי – Hebrew" lang="he" hreflang="he" data-title="שבר מצרי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Egyptische_breuk" title="Egyptische breuk – Dutch" lang="nl" hreflang="nl" data-title="Egyptische breuk" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%82%B8%E3%83%97%E3%83%88%E5%BC%8F%E5%88%86%E6%95%B0" title="エジプト式分数 – Japanese" lang="ja" hreflang="ja" data-title="エジプト式分数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Egyptisk_br%C3%B8k" title="Egyptisk brøk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Egyptisk brøk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/U%C5%82amek_egipski" title="Ułamek egipski – Polish" lang="pl" hreflang="pl" data-title="Ułamek egipski" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fra%C3%A7%C3%B5es_eg%C3%ADpcias" title="Frações egípcias – Portuguese" lang="pt" hreflang="pt" data-title="Frações egípcias" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%95%D0%B3%D0%B8%D0%BF%D0%B5%D1%82%D1%81%D0%BA%D0%B8%D0%B5_%D0%B4%D1%80%D0%BE%D0%B1%D0%B8" title="Египетские дроби – Russian" lang="ru" hreflang="ru" data-title="Египетские дроби" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%8A%E0%B6%A2%E0%B7%92%E0%B6%B4%E0%B7%8A%E0%B6%AD%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%E0%B6%B7%E0%B7%8F%E0%B6%9C%E0%B6%BA" title="ඊජිප්තියානු භාගය – Sinhala" lang="si" hreflang="si" data-title="ඊජිප්තියානු භාගය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Egyptian_fraction" title="Egyptian fraction – Simple English" lang="en-simple" hreflang="en-simple" data-title="Egyptian fraction" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Egip%C4%8Danski_ulomek" title="Egipčanski ulomek – Slovenian" lang="sl" hreflang="sl" data-title="Egipčanski ulomek" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Egyptil%C3%A4inen_murtoluku" title="Egyptiläinen murtoluku – Finnish" lang="fi" hreflang="fi" data-title="Egyptiläinen murtoluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%84%D0%B3%D0%B8%D0%BF%D0%B5%D1%82%D1%81%D1%8C%D0%BA%D0%B8%D0%B9_%D0%B4%D1%80%D1%96%D0%B1" title="Єгипетський дріб – Ukrainian" lang="uk" hreflang="uk" data-title="Єгипетський дріб" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_s%E1%BB%91_Ai_C%E1%BA%ADp" title="Phân số Ai Cập – Vietnamese" lang="vi" hreflang="vi" data-title="Phân số Ai Cập" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%A4%E5%9F%83%E5%8F%8A%E5%88%86%E6%95%B8" title="古埃及分數 – Chinese" lang="zh" hreflang="zh" data-title="古埃及分數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 28 October 2024, at 13:40 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About 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[\"Sfnp\"] = 14,\n [\"Sfrac\"] = 41,\n [\"Short description\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-qmvs7","timestamp":"20241122141243","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Egyptian fraction","url":"https:\/\/en.wikipedia.org\/wiki\/Egyptian_fraction","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1764362","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1764362","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-10-07T19:45:06Z","dateModified":"2024-10-28T13:40:31Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d9\/Rhind_Mathematical_Papyrus.jpg","headline":"finite sum of distinct unit fractions"}</script><script>(window.NORLQ=window.NORLQ||[]).push(function(){var ns,i,p,img;ns=document.getElementsByTagName('noscript');for(i=0;i<ns.length;i++){p=ns[i].nextSibling;if(p&&p.className&&p.className.indexOf('lazy-image-placeholder')>-1){img=document.createElement('img');img.setAttribute('src',p.getAttribute('data-src'));img.setAttribute('width',p.getAttribute('data-width'));img.setAttribute('height',p.getAttribute('data-height'));img.setAttribute('alt',p.getAttribute('data-alt'));p.parentNode.replaceChild(img,p);}}});</script> </body> </html>