Constructive representation of primitive Pythagorean triples

TL;DR

This paper introduces a systematic method for constructing and ordering primitive Pythagorean triples using geometric representations involving gnomons and arithmetic progressions, revealing new properties and mappings.

Contribution

It provides a novel geometric and algebraic framework for representing and enumerating primitive Pythagorean triples with explicit inverse mappings and properties.

Findings

Established a one-to-one correspondence between triples and gnomon partitions

Identified properties of arithmetic progressions linked to triples

Described the transition from primitive to general Pythagorean triples

Abstract

The paper presents a systematic construction of primitive Pythagorean triples. The order of enumeration on the set of primitive Pythagorean triples is defined. The order is based on the representation of a primitive Pythagorean triple by dividing the side of the generating square into two groups of factors using gnomons. In paper is shown the inverse mapping of the elements of a primitive Pythagorean triple to the parameters of the partition of the side of the generating square, the representation of primitive Pythagorean triples as the sum of two connected gnomons, a one-to-one correspondence of the connected gnomons of the primitive Pythagorean triple with the corresponding arithmetic progressions. Also the property of such arithmetic progressions as connectedness with each other, namely, partial overlap, is found. The motion (configuration change) of the connected gnomons is shown during the transition of a primitive Pythagorean triple to a general Pythagorean triple, all the terms of which have one common coefficient.