How many bases $b$ are there such that $663_b$ is prime?
663 base 10 =3023 base 6, which is a prime. That is all I can see up to base 20.
Considering the base $b$ representation of this number, it equals $6\cdot b^2 + 6 \cdot b + 3 = 3(2 \cdot b^2 + 2 \cdot b + 1)$. So for any choice of base $b,$ we see that $663_b$ is divisible by 3. Therefore there is no $b$ for which $663_b$ prime.