Given a prime $p$ of the form $4k+1$, is the integer right-angled triangle with hypotenuse $p$ unique?
I know Fermat's theorem says every prime of the form $4k+1$ can be uniquely written as the sum of two squares. And here we have to find whether the representation $(4k+1)^2=a^2+b^2$ is unique? I could not use the same proof here.
The primitive Pythagorean triples with $y$ even are parameterised by $$(x,y,z)=(r^2-s^2,2rs,r^2+s^2)$$ where $r$, $s$ are coprime, $r+s$ is odd, and $r>s>0$. In your case as the hypotenuse is prime, the triple must be primitive. Then by Fermat, $r$ and $s$ are uniquely determined.