[Math] Fermat’s theorem on sums of two squares

Question

There's Fermat's theorem on sums of two squares.

As the prime numbers that are $1\bmod4$ can be divided into the sum of two squares, will the squared numbers be unique?

For example, $41=4^2+5^2$ and the squared numbers will be $4$ and $5$.

Answer 1

Just to complement Pantelis' answer, the reason why they are unique can be easily seen from the proof using the Gaussian integers $\mathbb{Z}[i]$, which is a UFD.