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Fermat polygonal number theorem
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<h2 id="examples">Examples</h2> <p>Three such representations of the number 17, for example, are shown below: </p> <ul><li>17 = 10 + 6 + 1 (<i>triangular numbers</i>)</li> <li>17 = 16 + 1 (<i>square numbers</i>)</li> <li>17 = 12 + 5 (<i>pentagonal numbers</i>).</li></ul> <h2 id="history">History</h2> <p>The theorem is named after <a href="/facts/Pierre_de_Fermat/FLbUh75u">Pierre de Fermat</a>, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> <a href="/facts/Joseph_Louis_Lagrange/bTQUaSTv">Joseph Louis Lagrange</a> proved the <a href="/facts/Lagrange%2527s_four-square_theorem/ysLErgYw">square case</a> in 1770, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Gauss</a> proved the triangular case in 1796, commemorating the occasion by writing in <a href="/facts/Gauss%2527s_diary/0nDa4m6q">his diary</a> the line "<a href="/facts/Eureka_(word)/ekEvoR7D">ΕΥΡΗΚΑ!</a> num = Δ + Δ + Δ",<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> and published a proof in his book <a href="/facts/Disquisitiones_Arithmeticae/VQ38ReTh">Disquisitiones Arithmeticae</a>. For this reason, Gauss's result is sometimes known as the Eureka theorem.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The full polygonal number theorem was not resolved until it was finally proven by <a href="/facts/Cauchy/aoTjgj6j">Cauchy</a> in 1813.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The proof of Nathanson (1987) is based on the following lemma due to Cauchy: </p><p>For odd positive integers a and b such that <i>b</i>2 < 4<i>a</i> and 3<i>a</i> < <i>b</i>2 + 2<i>b</i> + 4 we can find nonnegative integers s, t, u, and v such that <i>a</i> = <i>s</i>2 + <i>t</i>2 + <i>u</i>2 + <i>v</i>2 and <i>b</i> = <i>s</i> + <i>t</i> + <i>u</i> + <i>v</i>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Pollock%2527s_conjectures/BpEdHtR6">Pollock's conjectures</a></li> <li><a href="/facts/Waring%2527s_problem/Qq5My8wy">Waring's problem</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html">"Fermat's Polygonal Number Theorem"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="/facts/Thomas_Little_Heath/E9xE228u">Heath, Sir Thomas Little</a> (1910), <a href="https://archive.org/details/diophantusofalex00heatiala"><i>Diophantus of Alexandria; a study in the history of Greek algebra</i></a>, Cambridge University Press, p. 188.</li> <li><a href="/facts/Melvyn_B._Nathanson/wPVRK8Ht">Nathanson, Melvyn B.</a> (1987), "A short proof of Cauchy's polygonal number theorem", <i>Proceedings of the American Mathematical Society</i>, 99 (1): 22–24, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2046263">10.2307/2046263</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2046263">2046263</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0866422">0866422</a>.</li> <li>Nathanson, Melvyn B. (1996), <i>Additive Number Theory The Classical Bases</i>, Berlin: <a href="/facts/Springer_Science%252BBusiness_Media/nAesf6nT">Springer</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94656-6. Has proofs of Lagrange's theorem and the polygonal number theorem.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Heath (1910). - Heath, Sir Thomas Little (1910), Diophantus of Alexandria; a study in the history of Greek algebra, Cambridge University Press, p. 188 <a href="https://archive.org/details/diophantusofalex00heatiala" target="_blank">https://archive.org/details/diophantusofalex00heatiala</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Heath (1910). - Heath, Sir Thomas Little (1910), Diophantus of Alexandria; a study in the history of Greek algebra, Cambridge University Press, p. 188 <a href="https://archive.org/details/diophantusofalex00heatiala" target="_blank">https://archive.org/details/diophantusofalex00heatiala</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Bell, Eric Temple (1956), "Gauss, the Prince of Mathematicians", in Newman, James R. (ed.), The World of Mathematics, vol. I, Simon & Schuster, pp. 295–339. Dover reprint, 2000, ISBN 0-486-41150-8. <a href="/wiki/Eric_Temple_Bell" target="_blank">/wiki/Eric_Temple_Bell</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Ono, Ken; Robins, Sinai; Wahl, Patrick T. (1995), "On the representation of integers as sums of triangular numbers", Aequationes Mathematicae, 50 (1–2): 73–94, doi:10.1007/BF01831114, MR 1336863, S2CID 122203472. <a href="/wiki/Aequationes_Mathematicae" target="_blank">/wiki/Aequationes_Mathematicae</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Heath (1910). - Heath, Sir Thomas Little (1910), Diophantus of Alexandria; a study in the history of Greek algebra, Cambridge University Press, p. 188 <a href="https://archive.org/details/diophantusofalex00heatiala" target="_blank">https://archive.org/details/diophantusofalex00heatiala</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>