It has been known for some 2000 years how to generate Pythagorean Triples. While the classical formulas generate all of the primitive triples, they do not generate all of the triples. For example, the triple (9, 12, 15) can’t be generated from the formulas, but it can be produced by introducing a multiplier to the primitive triple (3, 4, 5). And while the classical formulas produce the triple (3, 4, 5), they don’t produce the triple (4, 3, 5); a transposition is needed. This paper explores a new set of formulas that, in fact, do produce all of the triples i.e. every triple can be produced with a unique set of integer generators; there is no need for multipliers or transpositions. An unexpected result is an application to cryptology.
Spezeski, William J.
Rethinking Pythagorean Triples,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 3,
1, Article 9.
Available at: https://digitalcommons.pvamu.edu/aam/vol3/iss1/9