Random Sequences from Primitive Pythagorean Triples

This paper shows that the six classes of PPTs can be put into two groups. Autocorrelation and cross-correlation functions of the six classes derived from the gaps between each class type have been computed. It is shown that Classes A and D (in which the largest term is divisible by 5) are different from the other four classes in their randomness properties if they are ordered by the largest term. In the other two orderings each of the six random Baudhāyana sequences has excellent randomness properties. Introduction This article extends the results of a previous article on primitive Pythagorean triples (PPTs) [1], which presented their historical background and some advanced properties. This research is part of a program to use mathematical functions to generate random numbers [2]-[9]. There has been much recent research on families of Pythagorean triples [10]-[12]. A Pythagorean triple (a, b, c) is a set of integers that are the sides of a right triangle and thus a 2 + b 2 = c 2 . Given a Pythagorean triple (a, b, c), (da, db, dc) is also a triple. A primitive Pythagorean triple (PPT) consists of numbers that are relatively prime. To generate PPTs, one may start with different odd integers s and t that have no common factors and compute: 2 ; 2 ; 2 2 2 2 t s c t s b st a      There exist an infinity of PPTs. The coordinate (a/c,b/c) may be seen as a point on the unit circle, implying that a countably infinity of these points are rational. A sequence that generates a subset of PPTs is (2n+1, 2n 2 +2n, 2n 2 +2n+1) for n = 1,2,3... Indexing Using s and t Numbers For a convenient indexing one may use relatively prime s and t numbers in an array where s > t. This may be seen in the diagram shown below: