On two elliptic curves associated with perfect cuboids

Abstract

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all integer edges and diagonals, which is an old unsolved problem. Recently, based on a symmetry approach, it was shown that edges and face diagonals of rational perfect cuboid are roots of two cubic equations whose coefficients depend on two rational parameters. Six special cases where these cubic equations are reducible have been already found. Two more possible cases of reducibility for these cubic equations are considered in the present paper. They lead to a pair of elliptic curves.

AmSTeX, 11 pages, amsppt style

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Cited by in corpus (6)

• A note on rational and elliptic curves associated with the cuboid factor equations
• Asymptotic estimates for roots of the cuboid characteristic equation in the linear region
• A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture
• 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
• A fast modulo primes algorithm for searching perfect cuboids and its implementation