In reference to a prior question, I "think" that no $2$-or-more Pythagoean triples have the same product of $A\times B\times C$ but I don't know how to prove it. All primitive triples are unique in their values of ABC. There are an infinite number of these products. None appear to be the same. How do I prove that, for all combinations of $m,n$,
$2m_1n_1(m_1^2-n_1^2)(m_1^2+n_1^2)\ne 2m_2n_2(m_2^2-n_2^2)(m_2^2+n_2^2)?$
The pythagorean triples are integers and integers have a unique prime factorization.
Every primitive pythagorean triple contains 3 coprimes. If 2 triples had the same product, this would probably violate the prime factorization thrm.