Cuboid conjectures


Three cuboid conjectures are three mathematical propositions claiming irreducibility of three univariate polynomials with integer coefficients depending on several integer parameters. They are neither proved nor disproved.

The first cuboid conjecture

Cuboid conjecture 1. For any two positive coprime integer numbers
the eighth degree polynomial
is irreducible over the ring of integers .

The second cuboid conjecture

Cuboid conjecture 2. For any two positive coprime integer numbers
the tenth-degree polynomial
is irreducible over the ring of integers .

The third cuboid conjecture

Cuboid conjecture 3. For any three positive coprime integer numbers,, such that none of the conditions
is fulfilled the twelfth degree polynomial
is irreducible over the ring of integers .

Background

The conjectures 1, 2, and 3 are related to the perfect cuboid problem. Though they are not equivalent to the perfect cuboid problem, if all of these three conjectures are valid, then no perfect cuboids exist.