I’m revising for my upcoming exams and this is an unseen part from a previous exam:
In previous parts I’ve shown that the solutions to $x^2 +y^2=z^2$ are $$(x,y,z)=(a^2 -b^2,2ab,a^2+b^2)$$ and $$(x,y,z)=(2ab,a^2-b^2,a^2+b^2)$$ where $a$ and $b$ are positive integers. However the unseen part is, “Determine to what extent these integers $a$ and $b$ are uniquely determined”. Unfortunately I don’t even understand the question let alone how to attempt it. I’m not sure whether $a$ and $b$ are coprime, and if they are if that assumption helps at all. Any help explaining the question and giving a hint how to continue would be appreciated.