Perfect Cuboid and Congruent Number Equation Solutions
Abstract
A perfect cuboid (PC) is a rectangular parallelepiped with rational sides $a,b,c$ whose face diagonals $d_{ab}$, $d_{bc}$, $d_{ac}$ and space (body) diagonal $d_s$ are rationals. The existence or otherwise of PC is a problem known since at least the time of Leonhard Euler. This research establishes equivalent conditions of PC by nontrivial rational solutions $(X,Y)$} and $(Z,W)$} of congruent number equation $ y^2=x^3N^2x$, where product $XZ$ is a square. By using such pair of solutions five parametrizations of nearlyperfect cuboid (NPC) (only one face diagonal is irrational) and five equivalent conditions for PC were found. Each parametrization gives all possible NPC. For example, by using one of them  invariant parametrization for sides and diagonals of NPC are obtained: $a=2XZN$, $b=YW$, $c=XZ\sqrt{XZ}\,N$,$d_{bc}=XZN^2\sqrt{XZ}$, $d_{ac}=X+Z\sqrt{XZ}\,N$, $d_s=(XZ+N^2)\sqrt{XZ}$; and condition of the existence of PC is the rationality of $d_{ab} = \sqrt{Y^2W^2+4N^2X^2Z^2}$. Because each parametrization is complete, inverse problem is discussed. For given NPC is found corresponding congruent number equation (i.e. congruent number) and its solutions.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 arXiv:
 arXiv:1211.6548
 Bibcode:
 2012arXiv1211.6548M
 Keywords:

 Mathematics  Number Theory;
 11D25;
 11D41;
 11D72;
 14G05;
 14H52
 EPrint:
 27 pages