On the Pythagorean Triples’ Equations and the perfect cuboid problem.

Abstract

The perfect cuboid problem is a Pythagorean problem in nature. This paper gives several propositions regarding Pythagorean relationships and a discussion is made on the perfect cuboid problem. Among the proposi- tions is that the cuboid problem is a divisibility by 3 problem. Violation of the divisibility means that a perfect cuboid doesn’t exist for a given integer. Another consequence of divisibility by 3 and other theorems is that the fastest and most accurate algorithm for generating a random prime number p is, p = (5 × β) 2 + 1 2 where β = 2k + 1, k ∈ N.