I recently discovered what sexy primes are, primes of the form (p,p+6), where p is a prime. I have no background in number theory so pardon me if the answer is trivial.
So are there infinitely many pairs of them? Is there any reading material on them? Or a proof or counter example I can study? Thanks for your time.
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$\begingroup$We know that there is at least one difference smaller than seventy million (this result may have been improved since) which gives infinitely many pairs of primes. Whether $6$ is one of them we don't know.
$\endgroup$ 3 $\begingroup$There is lot of information about sexy primes here: . Even the largest sexy prime pair is stated as $11,593$ digits. But unfortunately nothing is said about finiteness of the pairs, triplets, etc.
$\endgroup$ 8 $\begingroup$See Polignac's conjecture, which asserts that for any even $n$ there are infinitely many prime $p$ for which $p + n$ is prime, i.e., there are infinitely many prime gaps of length $n$.
The strongest unconditional result in this direction is due to a Polymath wiki after Yitang Zhang's big breakthrough: there is some even $n \leq 246$ for which there are infinitely many prime gaps of length $n$.
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