#1**0 **

663 base 10 =3023 base 6, which is a prime. That is all I can see up to base 20.

Guest Mar 30, 2018

#2**+1 **

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.

supermanaccz Mar 30, 2018