If $a,b,c$ are primitive Pythagorean triples such that $c^2=a^2+b^2$, prove that this is unique primitive Pythagorean triple using $c$ as hypotenuse.
I'm not sure if I phrased this correctly. I want to ask, if $c^2=a^2+b^2$ where $a,b,c$ are pairwise coprime, prove that there are no $a_1$ and $b_1$ such that $a_1,b_1,c$ that satisfy $c^2=a_1^2+b_1^2$, and they are also Pythagorean triple.
This is obviously true based on the short list I found on Wiki but I can't seem to prove it... Help?
EDIT Since this obviously seems false, my question now is, if the process described above false only for $c=5k$? Since it seems to be false for $c=65, 145, 185, 205, 265...$ Can one prove this?