### Can we find Gaussian primes $\pi = 1 + 8 \mathbb{Z}[i]$ with $N(\pi) < 1000$?

It's an exercise in computational number theory. Either by hand or by computer, can we find the Gaussian primes $\pi = 1 + 8 \mathbb{Z}[i]$? To keep the list finite I guess we could have $N(\pi) < 10000$.

For example, is $\mathfrak{p} = (1+8i)$ a prime? Or $\mathfrak{p} = (-7 + 8i)$? I don't even know how to index the primes less than these. The norms are $1^2 + 8^2 = 65 = 5 \times 13$ and $(-7)^2 + 8^2 = 113$, so the first could factor and the second does not.

For $\mathfrak{p} = (a + bi)$ to check it is prime over $\mathbb{Z}[i]$, is it sufficient to check that $N(\mathfrak{p}) = a^2 + b^2$ is a prime over $\mathbb{Z}$?

It would be great to see the code in `Sage`

or `Pari/GP`

and that would be an acceptable answer.