# Math Help - Prime problem

1. ## Prime problem

Let p be prime with $p \equiv 1 (mod \ 4)$ Show that there are integers a,b for which $p=a^2 + b^2$

Let p be prime with $p \equiv 1 (mod \ 4)$ Show that there are integers a,b for which $p=a^2 + b^2$
By the other arguments $p$ is not irreducible. Thus, there exists $a,b,c,d\in \mathbb{Z}$ such that $p = (a+bi)(c+di)$ take the norm of both sides to get $p^2 = (a^2+b^2)(c^2+d^2)$. Now work with unique factorization over $\mathbb{Z}$. It cannot be that one of those factors is $1$ and the other $p^2$ because we cannot write $1=a^2+b^2$ if $a,b\not = 0$. Thus, it means $p=a^2+b^2=c^2+d^2$. (In fact, $a^2+b^2$, and $c^2+d^2$ are exactly the same, i.e. there is also uniqueness involved but that is a different story).