We're going to use the fact that:
So we then have:
This means that:
But is prime. So what can you conclude about its divisor ?
How should i start off this proof?
If (2^n -1) is prime, prove that n is prime.
So far I've only got 2 things written down:
1. If 2^n -1 is prime then using divisor notation. 1|(2^n -1) and (2^n -1)|(2^n -1). But I dont think this helps.
2. To prove this we can prove gcd((2^n -1),n) = 1 (i.e relatively prime)
Other than that not much idea at all.
Any suggestions would be greatly appreciated.
Since (2^n -1) is prime anything that divides it must be (2^n -1) or 1 itself so (2^a -1)|(2^n -1) implies that
(2^a -1) = (2^n -1) or (2^a -1)=1
=> a = n or => a = 1
so since n=ab then b = 1 or b = n.
Therfore if (2^n -1) is prime then n=ab=n x 1 so n is also prime.