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

Yes, if you don't take into account the order of the two numbers or $\pm$ sign in front of the numbers.

Answer 2

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.

Answer 3

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma.

Further, primes of the form $p=4n+3$, never have a decomposition into $2$ squares, proven in various ways here.