 # Cuboid conjectures

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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 a u the eighth degree polynomial

is irreducible over the ring of integers Z .

## The second cuboid conjecture

Cuboid conjecture 2. For any two positive coprime integer numbers p q the tenth-degree polynomial

is irreducible over the ring of integers Z .

## The third cuboid conjecture

Cuboid conjecture 3. For any three positive coprime integer numbers a , b , u such that none of the conditions

is fulfilled the twelfth degree polynomial

is irreducible over the ring of integers Z .

## 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.

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