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 Prime repunits (Posted on 2016-11-24)
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.

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 Solution Comment 5 of 5 |
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.

 Posted by Brian Smith on 2018-12-15 22:42:37

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