Prime or Composite? With proof

[Hasib]

Determine the following number is prime or composite \[2^{2^{2011}+2011}+1\]


HINTS: There is a generalize form of this problem.

[Moon]

$2^{2^{2011}+2011}+1 \equiv 0 \pmod{3}$

[Hasib]

hahaha. I am a grt fool. I dont check that, the power of 2 is odd. Thus the solution is so easy. By the way, the main problem is to prove $2^{2^k+s}+1$ is a composite number while $s \in \{1,2,3,4,...,(2^k-1)\}$.

So easy!! Try to do that $a^n+b^n$ is a composite number, and divisible by $a+b$ whence $n$ is $odd$

[Tahmid Hasan]

actually i was the first one to prove it and sakal roy generalized it

[Masum]

Let $s=2^lm$ with $m$ odd.Then we have $l<k$ and then $2^{2^k+s}+1=2^{2^l(2^{k-l}+m)}+1$ with $2^{k-l}+m$ odd