Residue Classes of the PPT Sequence

TL;DR

This paper explores the residue class structure of primitive Pythagorean triples, analyzes their autocorrelation properties, and demonstrates their potential use as pseudorandom sequences in cryptographic applications.

Contribution

It introduces a novel residue class classification for PPTs, derives probability results, and characterizes the autocorrelation properties of associated sequences, highlighting their randomness qualities.

Findings

Probability that the smaller odd number in PPT is divisible by p is 2/(p+1)

Autocorrelation function of Baudhayana sequences shows excellent randomness properties

Analytical explanation provided for autocorrelation peak and off-peak values

Abstract

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 Baudhayana 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.