Inverse problems associated with perfect cuboids

Abstract

A perfect cuboid is a rectangular parallelepiped with integer edges, integer face diagonals, and integer space diagonal. Such cuboids have not yet been found, but nor has their existence been disproved. Perfect cuboids are described by a certain system of Diophantine equations possessing an intrinsic $S_3$ symmetry. Recently these equations were factorized with respect to this $S_3$ symmetry and the factor equations were transformed into $E$-form. As appears, the transformed factor equations are explicitly solvable. Based on this solution, polynomial inverse problems are formulated in the present paper.

AmSTeX, 11 pages, amsppt style

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Cited by in corpus (11)

• 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
• A note on solutions of the cuboid factor equations
• On two elliptic curves associated with perfect cuboids
• 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