Question

Prove that if 2" + 1 is prime, then n is a power of 2.

Answer 1

A little bird told me that if 2n+12n+1 is prime, then nn is a power of 22. I tend not to trust talking birds, so I'm trying to verify that statement independently.

Suppose nn is not a power of 22. Then n=a⋅2mn=a⋅2m for some aa not a power of 22 and some integer mm. This gives 2n+1=2a⋅2m+12n+1=2a⋅2m+1. Now I suspect there's a way to factor that, but I don't see how. Can someone give me a hint?