On Pythagorean triplets

TL;DR

This paper explores properties of primitive Pythagorean triplets, revealing how their structure relates to prime factorization and modular constraints, and providing formulas to generate all solutions based on prime peaks.

Contribution

It introduces new relationships between prime factors of Pythagorean peaks and the structure of primitive triplets, including formulas for generating solutions from prime peaks.

Findings

Triplets with non-prime peaks derive from prime peak triplets.

Number of solutions relates to the prime factor count of the peak.

Peaks must be of the form 12k+1 or 12k+5, excluding certain residues.

Abstract

We discuss properties of diophantine solutions of the Pythagoras equation, $a^2+b^2=c^2$, where the three numbers have no common factor. Some of the highlights are: (1) All triplets for which $c$ (called the `peak') is non-prime can be deduced from the triplets with prime peaks; (2) If a peak has $n+1$ prime factors, there are $2^n$ independent solutions of the Pythagoras equation; (3) All Pythagorean peaks have to be of the form $12k+1$ or $12k+5$ for integer $k$; (4) A Pythagorean peak cannot have 3, or any number of the form $12k+7$ or $12k+11$, as its prime factors.