:::: '''[9, 12, 15] and [15, 20, 25] can be obtained just as easily from Euclid using ''m'' = 2, ''n'' = 1, ''k'' = 3 for the first one , and ''m'' = 2, ''n'' = 1, ''k'' = 45 for the second. This is clearly explained and sourced in the Wikipedia article on [[Pythagorean triple|Pythagorean triples]], along with an explanation for the the need to introduce parameter ''k'' when generating ALL triples as opposed to just the primitives. But as Euclid well knew, it is enough to consider only the set of ''primitive'' triples (all of which are generated by the equation you cite), since ALL ''non-primitive'' solutions can be generated trivially from the primitive ones. The version of the equation using only parameters ''m'' and ''n'' produces somean infinite number of non-primitive triples of the form [''ak'',''bk'',''ck''] where ''k'' is an "even square (or half-square) integer". To get only the primitives, ''m'' and ''n'' must be coprime, with ''m'' > ''n'', and one of ''m'',''n'' must be even.'''
