Prime or Composite? With proof

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Hasib
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Prime or Composite? With proof

Unread post by Hasib » Fri Dec 31, 2010 12:57 am

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


HINTS: There is a generalize form of this problem.
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Moon
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Re: Prime or Composite? With proof

Unread post by Moon » Sun Jan 02, 2011 3:32 am

$2^{2^{2011}+2011}+1 \equiv 0 \pmod{3}$
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Hasib
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Re: Prime or Composite? With proof

Unread post by Hasib » Sun Jan 02, 2011 6:08 pm

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 $odd$
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Tahmid Hasan
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Re: Prime or Composite? With proof

Unread post by Tahmid Hasan » Mon Jan 03, 2011 9:57 am

actually i was the first one to prove it and sakal roy generalized it
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Masum
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Re: Prime or Composite? With proof

Unread post by Masum » Thu Jan 13, 2011 12:20 pm

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
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