Research for a set of particular primitive Pythagorean triples

TL;DR

This paper explores a recurrence relation for a specific set of primitive Pythagorean triples where the difference between the two smaller numbers is 7, extending previous work that focused on a difference of 1.

Contribution

It introduces a new recurrence relation for primitive Pythagorean triples with a difference of 7, expanding the understanding of their structure beyond the case of difference 1.

Findings

Established a recurrence relation for triples with difference 7

Connected the sequence of terms to primitive Pythagorean triples

Extended Cimmino's previous results to a new difference case

Abstract

The set of terms of an infinite sequence expressed by a recurrence relation is equal to the set of maximum numbers of all primitive Pythagorean triples such that the difference between the two non-maximum numbers is 1, which Cimmino showed. By reference to Cimmino's researches, we show a recurrence relation for the case where the difference between the two non-maximum numbers is not 1, but 7.