that none exist. Exhaustive computer searches show that, if a perfect cuboid exists, the odd edge must be greater than 2.5 × 1013, the smallest edge must be greater than . the space diagonal must be greater than 9 × 1015. Some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists, based on modular arithmetic: One edge, two face diagonals and the space diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16. Two edges must have length divisible by 3 and at least one of those edges must have length divisible by 9. One edge must have length divisible by 5. One edge must have length divisible by 7. One edge must have length divisible by 11. One edge must have length divisible by 19. One edge or space diagonal must be divisible by 13. One edge, face diagonal or space diagonal must be divisible by 17. One edge, face diagonal or space diagonal must be divisible by 29. One edge, face diagonal or space diagonal must be divisible by 37. In addition: The space diagonal is neither a prime power nor a product of two primes. The space diagonal can only contain prime divisors ≡ 1(mod 4). If a perfect cuboid exists and are its edges, — the corresponding face diagonals and the space diagonal , then The triangle with the side lengths is a Heronian triangle an area with rational angle bisectors. The acute triangle with the side lengths , the obtuse triangles with the side lengths are Heronian triangles, with area equal to . Cuboid conjectures Three cuboid conjectures are three mathematical propositions claiming irreducibility of three univariate polynomials with integer coefficients depending on several integer parameters. The conjectures 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. They are neither proved nor disproved. Cuboid conjecture 1. For any two positive coprime integer numbers the eighth degree polynomial is irreducible over the ring of integers . Cuboid conjecture 2. For any two positive coprime integer numbers the tenth-degree polynomial is irreducible over the ring of integers . Cuboid conjecture 3. For any three positive coprime integer numbers , , such that none