On the Pythagorean Triples' Equations and the Perfect cuboid problem.

The perfect cuboid problem is a Pythagorean problem in nature. Thispaper gives several propositions regarding Pythagorean relationships anda discussion is made on the perfect cuboid problem. Among the proposi-tions is that the cuboid problem is a divisibility by 3 problem. Violationof the divisibility means that a perfect cuboid doesn’t exist for a giveninteger.Another consequence of divisibility by 3 and other theorems is thatthe fastest and most accurate algorithm for generating a random primenumber p is,p =(5 × β)2 + 12where β = 2k + 1, k ∈ N.