Perfect Cuboid and Congruent Number Equation Solutions

الأصل · EN

A perfect cuboid (PC) is a rectangular parallelepiped with rational sides a,b,c whose face diagonals dab, dbc, dac and space (body) diagonal dₛ 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 y²=x³-N²x, 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|√XZN,dbc=|XZ-N²|√XZ, dac=|X+Z|√XZN, dₛ=(XZ+N²)√XZ; and condition of the existence of PC is the rationality of dab = √Y²W²+4N²X²Z². 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.