Two and three descent for 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. Recently it was shown that the Diophantine equations describing such a cuboid lead to a couple of parametric families of elliptic curves. Two and three descent methods for calculating their ranks are discussed in the present paper. The elliptic curves in each parametric family are subdivided into two subsets admitting 2-descent and 3-descent methods respectively.

A fast modulo primes algorithm for searching perfect cuboids and its implementation

“…The first of them [52][53][54] continues the research on cuboid conjectures. The second one [55][56][57][58][59][60][61][62][63][64][65][66][67] relates perfect cuboids with multisymmetric polynomials.…”

A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture

“…The papers [48][49][50][51][52][53][54][55][56][57][58][59][60] deal with another approach based on so-called multisymmetric polynomials. In this paper we do not touch this approach.…”

A note on invertible quadratic transformations of the real plane

“…At the present time the case of the second cuboid conjecture is being intensively studied using the asymptotic approach (see [56][57][58][59]). The approach used in [60][61][62][63][64][65][66][67][68][69][70][71][72] is quite different. We do not consider this approach below in the present paper.…”