A Better Proof

One of any three consecutive numbers must be divisible by 3.

2^n-1, 2^n, 2^n+1 are three consecutive numbers.

2^n-1 is not divisible by 3 because it it prime.

2^n is not divisible by 3 because it is divisible by 2 and its powers.

Then 2^n+1 must be divisible by 3.

Therefore, 2^n+1 is composite.