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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.2012-09-27
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    ...and the other one involving predecessors to powers of two is about [Mersenne primes](http://en.wikipedia.org/wiki/Mersenne_prime), $2^p-1$. The largest known primes are constructed this way...2012-09-27

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