Need hints for a problem about primes.

From: Snis Pilbor (snispilbor_at_yahoo.com)
Date: 01/25/05 

Date: 24 Jan 2005 21:22:48 -0800

Hello,

I am clawing my hair out trying to figure this out. The problem
is to prove that if a^n+1 is prime (a>1) then n is a power of 2. I
honestly don't even know where to start. Certainly this would imply
a^n == -1 (p) [where prime p is just a^n+1, but we "forget" this
relationship with a and just focus on the fact that a^n is -1 mod 'some
prime'], and this is the only starting point I can think of, but this
is a long long way away from a solution and I'm hopelessly stumped of
even what broad, general strategy to use. Other grasps in the dark
include observing that a^2n == 1 (p), and if we can show 2n is a power
of 2 we're done, but this too leads nowhere; and that we can safely
assume a is itself not a nontrivial power of anything (or just pull
that power into n). Obviously a is even.

Fair warning: this problem DOES arise amid homework. I'm just
hoping for a hint or clue...

     Snis Pilbor

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