http://openproblemgarden.org/op/perfect_cuboid Comments for "Perfect cuboid" en http://openproblemgarden.org/op/perfect_cuboid#comment-7133 <p>send to tomk@globalserve.net</p> Fri, 27 Jan 2012 14:53:23 +0100 Anonymous comment 7133 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid#comment-6958 <p>I found the divisibility conditions of four sides a, b, c and d in a primitive 4d euler brick (if exists):<br /> 1. One is divided by 64, another by 16, another by 4, another odd.<br /> 2. One is divided by 27, another by 9, another by 3, another not by 3.<br /> 3. Two is divided by 5.<br /> 4. Two is divided by 11.<br /> 5. One is divided by 13.<br /> 6. One is divided by 19.</p> Sun, 24 Apr 2011 01:31:11 +0200 tsihonglau comment 6958 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid#comment-6884 <p>If you still think you have a proof, I'd love to take a look - my email is timro21@gmail.com, or you could have it published on the Unsolved Problems web site at http://www.unsolvedproblems.org/</p> <p>Tim</p> Mon, 27 Dec 2010 23:50:45 +0100 Anonymous comment 6884 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid#comment-6822 <p>Without loss of generality, we can suppose a > b > c > d and remove a Diophantine equation from the system. I found some solutions, for example: without the last equation, the following quadruple is a solution. a=6325,b=5796,c=5520,d=528. They are so small, so i guess 4D euler bricks should exist.</p> Fri, 01 Oct 2010 14:59:58 +0200 tsihonglau comment 6822 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid#comment-6782 <p>I think that I have a simple proof that there cannot be any primitive perfect cuboid (primitive perfect Euler brick). I am willing to provide it if anyone requests it. T Herndon</p> Tue, 24 Aug 2010 23:45:48 +0200 Anonymous comment 6782 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid#comment-6734 <p>Well, it's easy to show that for any primitive 4D brick (that is, one where a, b, c, and d have no common factor), then exactly one of a, b, c, and d must be odd, and the rest even.... </p> Wed, 19 May 2010 02:04:04 +0200 Anonymous comment 6734 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid#comment-6726 <p>Perfect cuboid is related to Euler brick whose edges and face diagonals are all integers. It is know that there are infinite Euler bricks. But is there any 4D Euler brick? In other words, is there any solution to the following system of Diophantine equations:</p> <p><img class="teximage" src="/files/tex/ac50f2b5cf0ec84c40d571fc04650b820cccac3a.png" alt="$ a^2 + b^2 = e^2 $" /></p> <p><img class="teximage" src="/files/tex/0c16c77fce704820a79bfa1bcef5efc106005450.png" alt="$ a^2 + c^2 = f^2 $" /></p> <p><img class="teximage" src="/files/tex/9905f90ff9377dfbf429a1849913b66391be03a4.png" alt="$ b^2 + c^2 = g^2 $" /></p> <p><img class="teximage" src="/files/tex/b1ebed5319923ae22e3a250eddf8348a2a79c2f3.png" alt="$ a^2 + d^2 = h^2 $" /></p> <p><img class="teximage" src="/files/tex/c654a6fe2e73072927c6128611ec8fd48eb36010.png" alt="$ b^2 + d^2 = i^2 $" /></p> <p><img class="teximage" src="/files/tex/f779281032ac14a3b2c227e5fb029817a438185b.png" alt="$ c^2 + d^2 = j^2 $" /></p> <p>I computed a, b, c, d up to 1 million with brute force and found no solution. Any idea?</p> Tue, 04 May 2010 16:17:10 +0200 tsihonglau comment 6726 at http://openproblemgarden.org http://openproblemgarden.org/op/perfect_cuboid <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> » <a href="/category/computational_number_theory">Computational N.T.</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Does a perfect cuboid exist?</p> </div> </tr></td> </table> </td> </tr> </table> Number Theory Computational Number Theory http://openproblemgarden.org/op/perfect_cuboid#comment Tue, 04 May 2010 15:57:14 +0200 tsihonglau 37221 at http://openproblemgarden.org