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 $y^2=x^3-N^{2}x$, where product $XZ$ is a square. By using such pair of solutions five parametrizations of nearly-perfect 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: $a=2XZN$, $b=|YW|$, $c=|X-Z|\sqrt{XZ}N$,$d_{bc}=|XZ-N^2|\sqrt{XZ}$, $d_{ac}=|X+Z|\sqrt{XZ}N$, $d_s=(XZ+N^2)\sqrt{XZ}$; and condition of the existence of PC is the rationality of $d_{ab}=\sqrt{Y^2{W}^2+4N^2{X}^2{Z}^2}$. 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.

27 pages

References in corpus (1)

• A note on rational and elliptic curves associated with the cuboid factor equations

Cited by in corpus (2)

• Parametric Solutions for a Nearly-Perfect Cuboid
• Diophantine Equations and Congruent Number Equation Solutions