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## Homework Statement

consider the number m=111...1 with n digits, all ones. Prove that if m is Prime, then n is prime

## Homework Equations

def of congruence. fermat's and euler's theorem. can also use σ(n): the sum of all the positive divisors of n, d(n): the number of positive divisors of n, φ(n): Euler's totient function φ, counting the positive integers coprime to (but not bigger than) n

## The Attempt at a Solution

I managed to prove that every odd prime except 5 divides on of these numbers.

3|111 so lets look at P>5 which do not divide 10.

now all such numbers can be represented by [tex]\frac{10^(^P^-^1^)-1}{9}[/tex] using fermat's theorem I can say this is congruent to 0(mod P). so now every prime P that i plug into this will give me a one of these numbers. obviously it doesn't work with non primes since we used fermat's little theorem.

so now i don't know what to do exactly.... if m is prime then i know it's not one of these numbers and the sum of it's digits is odd. but i am stuck.