A biquadratic Diophantine equation associated with perfect cuboids

Abstract

A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids are described by a system of four Diophantine equation possessing a natural $S_3$ symmetry. Recently these equations were factorized with respect to this $S_3$ symmetry and the factor equations were derived. In the present paper the factor equations are transformed to $E$-form and then reduced to a single biquadratic equation.

AmSTeX, 17 pages, amsppt style

References in corpus (4)

• A note on the third cuboid conjecture. Part I
• Perfect cuboids and multisymmetric polynomials
• On an ideal of multisymmetric polynomials associated with perfect cuboids
• On the equivalence of cuboid equations and their factor equations

Cited by in corpus (8)

• A general rational solution of an equation associated with perfect cuboids
• Inverse problems associated with perfect cuboids
• On singularities of the inverse problems associated with perfect cuboids
• A note on rational and elliptic curves associated with the cuboid factor equations
• On two algebraic parametrizations for rational solutions of the cuboid equations
• On two elliptic curves associated with perfect cuboids
• A note on solutions of the cuboid factor equations
• On a pair of cubic equations associated with perfect cuboids