Can two Pythagorean triples have the same product?

Question

I'm writing a program to search for Pythagorean triples, and I want to decrease the search-time as much as possible (obviously).

There's a method that will decrease the search-time drastically, if only it could be proven that no two Pythagorean triples will have the exact product of multiplying their terms.

For example, [3, 4, 5] and [15, 20, 25]. Indeed, 3 x 4 x 5 is not equal to 15 x 20 x 25. But could this be proven for all Pythagorean triples?

Answer 1

According to the comments of the OEIS sequence listing all these products, this is an open problem.

Answer 2

Given $\qquad A=m^2-n^2\qquad B=2mn\qquad C=m^2+n^2\qquad$ the product is $$P=(m^2-n^2)\times2mn\times(m^2+n^2)\quad=2m^5n-2mn^5\quad=2mn(m-n)(m+n)(m^2+n^2)$$

The fundamental theorem of arithmetic states that every integer greater than $1$ is either prime or the unique product prime factors. Therefore, no $2$ Pythagorean triples can have the same product.