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I'm reading up on prime pairs, and I had a question... I can't seem to find an answer to this anywhere, and the wikipedia list of prime types is enormous! Afraid I missed it when going through it.

I know that many primes come in twin, cousin or sexy pairs (or sexy triplets, etc). Where can I find a list of primes that are NOT 2, 4, or 6 units higher/lower than another prime? :) Is it called anything specific?

Thanks!

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closed as unclear what you're asking by Shailesh, Claude Leibovici, user91500, user223391, Watson Jul 22 '16 at 9:28

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These are just typical primes. No reason to give them a name because the majority of primes are of this "form" (and it's not so much a form as it is the absence of a form). Upper bound sieve theory shows that the number of primes up to $x$ which are twin/cousin/sexy is at most a constant times $\frac{x}{(\log x)^2}$, and it is conjectured that this is actually also the correct lower bound. Meanwhile, the number of primes up to $x$ is asymptotic to $\frac{x}{\log x}$, so the primes described in your question constitute a fraction of $1 -\frac{C}{\log x}$ of all primes. As $x$ grows very large this fraction gradually approaches 100% of primes; there is really nothing notable about these primes.

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  • $\begingroup$ Thanks! Do the gaps between primes follow any simple rules, or is it completely random? $\endgroup$ – Pepe Jul 22 '16 at 10:54
  • $\begingroup$ @Pepe No simple rules. The statistics of gaps are predicted to follow a certain distribution, and empirically they do, but they are still quite poorly understood from the viewpoint of what's proven. There have been huge breakthroughs in recent years via the works of Zhang, Maynard, Tao and others, yet we still are far from proving there are infinitely many twin primes. Try arxiv.org/abs/1102.0481 for some detailed heuristics. $\endgroup$ – Erick Wong Jul 22 '16 at 17:36

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