A number like 11, 111, or 1111 i.e. a number containing only the digit 1
is called
repunit.
List all base-8 prime repunits.
A base 8 repunit of x digits can be represented (in base 10) as 8^(x-1) + 8^(x-2) + ... + 8^2 + 8 + 1
The expression can also be expressed as (8^x-1)/7. An alternate factorization of the numerator is (2^x - 1) * (4^x + 2^x + 1); one of these two factors is divisible by 7.
For any x greater than 3 both (2^x - 1) * (4^x + 2^x + 1) are larger than 7, so the value will still be composite after 7 is divided out.
This leaves only three possible candidates for primes: 1, 11, or 111 base 8, which are 1, 9, and 73 base 10.
Then the only base 8 prime repunit is 111 = 73 base 10.
Solution
