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With initial observations, I would like to ask the following question:

Are there infinitely many primes of the form $2^{2^n}-1$ $(n\in \mathbb{N})$?

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The famous unsolved problem is about $2^{2^n}+1$ (Fermat numbers). It is not known whether there are infinitely many primes of this form. Indeed, only five of them are known, and there may be no more. –  André Nicolas Sep 27 '12 at 7:11
    
...and the other one involving predecessors to powers of two is about Mersenne primes, $2^p-1$. The largest known primes are constructed this way... –  Tobias Kienzler Sep 27 '12 at 9:36

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up vote 9 down vote accepted

Hint: $2^{2^n}-1=(2^{2^{n-1}}-1)(2^{2^{n-1}}+1)$

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