How many bases $b$ are there such that $663_b$ is prime?
Let "b" stand for any base.
663 = (6 x b2) + (6 x b) + 3
the above can be written 6b2 + 6b + 3
you can always factor a 3 out of 6b2 + 6b + 3
so, there is no base in which 663 is prime
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