#1**+1 **

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.

Guest Jun 30, 2019

#2**+1 **

How many bases *b* are there such that 663_{b} 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.

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Guest Jul 1, 2019

#4**+1 **

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.

Alan
Jul 1, 2019

#5**+1 **

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.

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Guest Jul 1, 2019