TITLE: Fermat's theorem on sums of two squares
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: 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.