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Does anyone know whether there are only finitely many of primes of the form $6^{2n}+1$, where $n$ zero or any natural number?

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    Any prime you pick that is of the form $6^{2n}+1$ will be finite, unless you're using definitions other than the most conventional ones. Maybe you meant to ask whether there are only finitely many of them? If so, that's the right way to say it. This is actually a fairly frequently occurring way of misunderstanding the terminology.2011-12-30
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    Thanks, Michael, for your suggestion.2011-12-30
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    Using the admittedly very bad heuristic assumption that $6^{2n}+1$ is randomly distributed, I would conjecture so on the basis of the Prime Number Theorem: the probability that $6^{2n}+1$ is prime is roughly $1/(\log 6^{2n})\sim 1/n$, and the harmonic series diverges giving an infinite expected value for the number of such primes.2011-12-30
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    After checking the values with wolfram alpha, it seems that it begins with a few primes but then each of the terms seem to be divisible by some previous term in the sequence... I would conjecture there are only finitely many of them, probably because of some simple algebra trick I can't quite think of.2011-12-30
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    @AlexBecker Maybe I am misinterpreting your argument, but doesn't that same heurestic tell us that there are infinitely many primes of the form $6^{2n}$?2011-12-30
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    @ Pedro.Yes, I checked to find that as well.$6^{6}+1|6^{12}+1$2011-12-30
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    Your numbers look to be special cases of [Pierpont primes](https://en.wikipedia.org/wiki/Pierpont_prime). It is conjectured that there are infinitely many of them.2011-12-30
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    @AlexYoucis: Yes, his heuristic does say that as well. But that's why it's a heuristic - we know they aren't actually randomly distributed. And for, $6^{2n}$ we understand the distribution easier.2011-12-30
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    @AlexYoucis Yes, that's why I said "very bad". However, liked mixedmath said, it at least isn't as clearly bad as applying the same heuristic to $6^{2n}$ which is definitely less randomly distributed for the purposes of the argument.2011-12-30

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