Thue’s lemma

Let $p$ be a prime number of the form $4k+1$ . Then there are two unique integers $a$ and $b$ with such that $p=a^2+b^2$. Additionally, if a number $p$ can be written in as the sum of two squares in 2 different ways (i.e. $p=a^2+b^2$ and $p=c^2+d^2$ with the two sums being different), then the number $p$ is composite.