A note on a perfect Euler cuboid

Abstract

The problem of constructing a perfect Euler cuboid is reduced to a single Diophantine equation of the degree 12.

Perfect cuboids and irreducible polynomials

“…The formulas (3.4), (3.5), (3.6) and (3.7) are taken from [35]. They can be verified by means of direct calculations.…”

“…It is easy to see that each rational Euler cuboid can be transformed to an Euler cuboid with integer sides and diagonals. In the case of perfect cuboids (either integer or rational) each such cuboid can be transformed to a perfect rational cuboid whose space diagonal is equal to unity (see [35]). Conversely each perfect rational cuboid with unit space diagonal yields some perfect cuboid with integer sides and diagonals.…”

“…In [35] the problem of finding a perfect cuboid was reduced to the following Diophantine equation of the order 12 with four variables a, b, c, and u: The exact result of the paper [35] is formulated as follows. A more simple equation associated with perfect cuboids was derived in [18] (see also [27]).…”

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

“…Theorem 1.1 can be found in [1]. It stems from the results of [2] and [3]. As for the perfect cuboid problem itself, it has a long history reflected in .…”

A note on the first cuboid conjecture

“…In [36] the problem of constructing perfect cuboids was reduced to the polynomial Diophantine equation P abu (t) = 0, where P abu (t) is given by the formula P abu (t) = t 12 + (6 u 2 − 2 a 2 − 2 b 2 ) t 10 + (u 4 + b 4 + a 4 + 4 a 2 u 2 + + 4 b 2 u 2 − 12 b 2 a 2 ) t 8 + (6…”