Unique primitive pythagorean triples for every integer, for every set of two integers, and for every set of three integers, one of which is a prime

Euclid presents formulae to generate primitive Pythagorean triples. The formula given by Euclid requires an input of two parameters, herein called M and N. Euclid’s formula has the “advantages” of (1) being unique, meaning any change in either M or N will change the triple that results; and (2) being exhaustive, meaning all primitive Pythagorean triples may be generated by Euclid’s formula. However, it has the “disadvantage” that there is a dependency between M and N, and they cannot be chosen independently. Davis, Appel and Seff (2019) (henceforth DAS) previously presented two alternate formulae to generate primitive Pythagorean triples that do not have this disadvantage. The first formula only requires one parameter which may be any positive integer, and the second requires an input of two parameters that are totally independent, and may be any positive integers. These two formulae are also unique, but they are not exhaustive—there exist primitive Pythagorean triples that are not generated by either the first or second of these formulae. In this paper we extend their work by presenting a third formula which requires an input of three parameters that are totally independent, and may be any positive integers, except that one of the parameters must be a prime number.