If 2n+12^n+1 is prime, why must nn be a power of 22?

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

Suppose n is not a power of 2. Then n=a2m for some a not a power of 2 and some integer m. This gives 2n+1=2a2m+1. Now I suspect there’s a way to factor that, but I don’t see how. Can someone give me a hint?

Answer

Hint. For any odd natural number a, the polynomial x+1 divides xa+1 evenly.

In particular, we have xa+1x+1=(x)a1(x)1=1x+x2+(x)a1 by the geometric sum formula. In this case, specialize to x=22m and we have a nontrivial divisor.

(Also, x^a+1\equiv(-1)^a+1\equiv-1+1\equiv0\mod x+1 inside \Bbb Z[ x] is pretty slick.)

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Source : Link , Question Author : Rob , Answer Author : Community

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