But all such triples are non-primitive. I.e. they are of the form (ka, kb, kc) with k>1. So all Pythagorean triples can be generated as a scaled primitive triple (a,b,c) that is generated by Euclid's formula from m,n.

Learned something new today. For other interested: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_...