The largest known prime number and Gauss.

The largest known prime number (As of March 2018) is $p = 2^{77232917} - 1$ .

True or false?

a) The diophantine equation $p = a^2 + b^2$ with $a, b \in \mathbb{Z}$ ( with $a, b$ being integers numbers), has at least a solution.

b) $p$ is prime and/or irreducible in $\mathbb{Z}[i]$ , the ring of Gaussian integers.