Residue Classes of the PPT Sequence

Primitive Pythagorean triples (PPT) may be put into different equivalence classes using residues with respect to primes. We show that the probability that the smaller odd number associated with the PPT triple is divisible by prime p is 2/(p+1). We have determined the autocorrelation function of the Baudhāyana sequences obtained from the residue classes and we show these sequences have excellent randomness properties. We provide analytical explanation for the peak and the average off-peak values for the autocorrelation function. These sequences can be used specifically in a variety of key generation and distribution problems and, more generally, as pseudorandom sequences. Introduction The theory of primitive Pythagorean triples (PPTs) [1],[2],[3] is related to other number theory areas such as Gaussian numbers, modular forms and spinors [4]. A Pythagorean triple (a, b, c) is a set of integers that are the sides of a right triangle and thus a2 + b2 = c2. In this triple a is the smaller of the two odd numbers in the 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 + = − = = In earlier papers [2],[3] we considered properties of PPTs that had been put into 6 classes based on divisibility by 3, 4, and 5. Once the sequence of the PPTs has been listed in terms of the classes, one can determine the separation between elements of the same class and thus obtain a numerical sequence. We have named these sequences Baudhāyana sequences [3] after the ancient mathematician who described Pythagorean triples before Pythagoras [5],[6]. (For those not familiar with the historical context, please see [7]-[10].) Here we consider properties of residue classes of the PPT sequence. We show that the probability that a PPT is divisible by prime p is 2/(p+1). We provide an analytical explanation for the general characteristics of the autocorrelation function and show that just like the sequences in 6 classes that were considered in [2], these sequences have excellent randomness properties.