A square + square =prime

We can use Fermat's theorem on sums of two squares-

If a prime $p$ can be expressed as $p=a^{2}+b^{2}$ , where $a$ and $b$ are positive integers, then $p\equiv 1 \pmod {4}$

The smallest such possible $p$ is $p=229$ . By trial and error, we have $a=2$ and $b=15$ , so $a+b=17$ .