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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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closed as unclear what you're asking by Lord Shark the Unknown, Professor Vector, Aqua, Dietrich Burde, kjetil b halvorsen Dec 20 '17 at 20:58

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ If you just recently discovered what sexy primes are, don't think everybody else knows, but include the definition, please. $\endgroup$ – Professor Vector Dec 20 '17 at 18:41
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    $\begingroup$ Why the motion to close this question? I am requesting references for it? Is that not a valid enough question? $\endgroup$ – user264750 Dec 20 '17 at 18:44
  • $\begingroup$ Voters of closing this question, please suggest improvements? I'm not sure why the question is unclear. $\endgroup$ – user264750 Dec 20 '17 at 19:21
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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.

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  • $\begingroup$ Is there a book or article on the asymptotic density of sexy primes you could recommend? $\endgroup$ – user264750 Dec 20 '17 at 18:39
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    $\begingroup$ @Hades All I know about the subject is more or less summarised in that video. Sorry. $\endgroup$ – Arthur Dec 20 '17 at 18:47
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    $\begingroup$ See the article Prime Gap in Wikipedia, especially the section Upper Bounds. The value 70 million (proved in 2013) is the first known bound but has since been improved considerably.. $\endgroup$ – DanielWainfleet Dec 20 '17 at 19:02
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There is lot of information about sexy primes here: https://en.wikipedia.org/wiki/Sexy_prime. Even the largest sexy prime pair is stated as $11,593$ digits. But unfortunately nothing is said about finiteness of the pairs, triplets, etc.

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    $\begingroup$ I'm not sure how knowing the largest pair tells us anything about their finiteness? Every year, we find a larger and larger prime, but we know there are infinitely many of them. $\endgroup$ – user264750 Dec 20 '17 at 18:28
  • $\begingroup$ What I mean was, when whether there are finitely or infinitely many pairs is not proved, knowing the largest pair has nothing to do with determining the finiteness. But your example was really nice, I appreciate it. $\endgroup$ – ArsenBerk Dec 20 '17 at 18:34
  • $\begingroup$ @Hades Such questions don't have useful answers. If you don't know an unsolved problem, don't complain if people are pointing that out. $\endgroup$ – Professor Vector Dec 20 '17 at 18:44
  • $\begingroup$ Sorry. I don't mean to be rude. But stating information from Wikipedia doesn't help. I am looking for mathematical texts. $\endgroup$ – user264750 Dec 20 '17 at 18:47
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    $\begingroup$ @Hades But you are rude. If you emphasize your non-existing background in number theory, how is anybody to know that Wikipedia isn't too much of a challenge for you, let alone articles about number theory? $\endgroup$ – Professor Vector Dec 20 '17 at 18:52
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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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