Let $p\in \mathbb{N}$ be a prime. Show that if $p$ is not the sum of two squares then $p$ is prime in $\mathbb{Z}[i]$
My approach was to assume $p$ is not a prime in $\mathbb{Z}[i]$. Then $\exists (s+it), (u+iv)$ such that $$p=(s+it)(u+iv)=(su-tv)+i(tu+sv).$$
Since $p$ is a natural prime number we have $tu+sv = 0$; therefore, $p=su-tv$.
I want to write $p$ in such way that it's the sum of two squares but I'm unsure how to proceed. Any suggestions?