A note on the third cuboid conjecture. Part I

Abstract

The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain Diophantine equation of the order 12. In this paper a structural theorem for the solutions of this Diophantine equation is proved.

AmSTeX, 34 pages, amsppt style

References in corpus (4)

• A note on the first cuboid conjecture
• Perfect cuboids and irreducible polynomials
• A note on a perfect Euler cuboid
• A note on the second cuboid conjecture. Part I

Cited by in corpus (16)

• Perfect cuboids and multisymmetric polynomials
• On an ideal of multisymmetric polynomials associated with perfect cuboids
• A biquadratic Diophantine equation associated with perfect cuboids
• On the equivalence of cuboid equations and their factor equations
• 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
• A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture
• Asymptotic estimates for roots of the cuboid characteristic equation in the linear region
• Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture
• A note on invertible quadratic transformations of the real plane
• On a pair of cubic equations associated with perfect cuboids
• A fast modulo primes algorithm for searching perfect cuboids and its implementation