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Prime or Composite? With proof
Posted: Fri Dec 31, 2010 12:57 am
by Hasib
Determine the following number is prime or composite \[2^{2^{2011}+2011}+1\]
HINTS: There is a generalize form of this problem.
Re: Prime or Composite? With proof
Posted: Sun Jan 02, 2011 3:32 am
by Moon
Re: Prime or Composite? With proof
Posted: Sun Jan 02, 2011 6:08 pm
by 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 $s \in \{1,2,3,4,...,(2^k-1)\}$.
Re: Prime or Composite? With proof
Posted: Mon Jan 03, 2011 9:57 am
by Tahmid Hasan
actually i was the first one to prove it and sakal roy generalized it
Re: Prime or Composite? With proof
Posted: Thu Jan 13, 2011 12:20 pm
by 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