number theory - Generalized Pythagorean triples construction.

All primitive Pythagorean triples $(a, b, c) : \{ a^2 + b^2 = c^2 \} \wedge \{ a \equiv 0 \pmod{2} \}$ can be expressed in the form: $$\{ a = 2pq, b = p^2 - q^2, c = p^2 + q^2 \}$$ for positive integers $p, q : \{ \gcd(p,q) = 1 \} \wedge \{ p \not\equiv q \pmod{2} \}$.

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I conjectured that this also holds for imprimitive Pythagorean triples (in this case $p,q$ are not necessarily relatively prime and of opposite parity).

However, I could not find any counterexamples and currently I am stuck in the developing of a proof.

That is why I am appealing to you. I would really appreciate any counterexamples, proofs, ideas, etc.

Thank you.

Answer

Sadly, it's not true for the general case. Easiest counterexample is to take the $3-4-5$ right triangle and multiply each side by $3$. $15$ cannot be written as the sum of $2$ squares. The sum of $2$ squares cannot be congruent to $3\pmod4$.