How many bases \(b\) are there such that 663\(_b\) is prime?
Here is my "guess" (if I understand your question) !.
I don't think there is any base between 7 and 36 where 663 is prime. I couldn't find a single base between 7 and 36 where 663 is a prime number. Sorry, that is the best I could do.
How many bases b are there such that 663b is prime?
I don't think there are any. 663 is the same quantity, irrespective of what base you write it in, and that quantity is divisible.
Base 2 can only have digits 0 and 1. No base up to and including 6 can have the digit 6 - that's why Guest #1 started at base 7.
tommarvoloriddle, my answer works for base 2
in base 10 -- 663 divided by 39 equals 17 therefore not prime
in base 2 -- 1010010111 divided by 100111 equals 10001 therefore not prime
That's what i meant by 663 is the same quantity no matter what base you write it in. My interpretation of the question was different from that of Guest #1. On reflection, and especially Alan's and your comments, I now realize I was looking at it the wrong way.