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Twin, cousin, and sexy primes are of the forms $(p,p+2)$, $(p,p+4)$, $(p,p+6)$ respectively, for $p$ a prime. The Wikipedia article on cousin primes says that, "It follows from the first Hardy–Littlewood conjecture that cousin primes have the same asymptotic density as twin primes," but the analogous article on sexy primes does not make a similar claim.

Q1. Are the sexy primes expected to have the same density as twin primes?

Q2. Is it conjectured that there are an infinite number of cousin and sexy prime pairs?

Q3. Have prime pairs of the form $(p,p+2k)$ been studied for $k>3$? If so, what are the conjectures?

Thanks for information or pointers!

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    There is a very general conjecture in the theory of prime numbers -- Schinzel's Hypothesis H (http://en.wikipedia.org/wiki/Schinzel's_hypothesis_H) which covers your Q2.2010-12-15
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    @Douglas: Exactly the type of info I seek. Thanks!2010-12-15
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    In regards to **Q2.** it has been proven that there are infinitely many primes with a gap of $6$. The gap was originally $63,374,611$ (rounded off to $70,000,000$) and then brought down over the years, especially by Terence Tao and the open Polymath project he launched in $2013$. Since every pair of sexy primes differ by $6$, then there are infinitely many sexy primes. The question still remains for cousin primes and twin primes.2018-02-14
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    @user477343 Can you give a reference for this? To Wikipedia's knowledge, the Polymath-reduction to 6 depends on the Elliott–Halberstam conjecture, which is not proven up to now. Your comment seems to imply, that there are indeed infinitely many sexy primes.2018-08-15
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    @Babelfish There is a booked called *Things to Make and Do in the Fourth Dimension* (2014) written by Australian stand-up comedian and mathematician (and member of the [Numberphile](https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A) crew) who talked about the polymath project in his book (Chapter Seven: *Prime Time*, page $152$). He wrote that as of $20$ July, $2013$, the brought the gap from just under "[$70$ Million](https://www.youtube.com/watch?v=vkMXdShDdtY)" to $5,414$. I then saw on wikipedia that they proved there are infinitely many primes with a gap of $246$.2018-08-15
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    @Babelfish I soon realised the book was outdated, and then Yitang Zhang proved that there exists an $n\leqslant 246$ such that there are infinitely many primes gaps of $n$ (see [here](https://math.stackexchange.com/questions/2574781/infinitely-many-pairs-of-sexy-primes)). I then stumbled upon wikipedia, that ***if*** the [Riemann Hypothesis](https://www.youtube.com/watch?v=d6c6uIyieoo) is true, then there are infinitely many primes with a gap of $6$. This, although a conjecture, is *highly* likely to be true; but the *only* evidence not agreeing is "the largest sexy prime" thus far (May 2009).2018-08-15
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    @Babelfish you can also go [here](https://math.stackexchange.com/questions/2880738/is-there-any-conjecture-that-has-been-proved-to-be-solvable-provable-but-whose-d) for the "$n\leqslant 246$" part (particularly [@GerryMyerson's answer](https://math.stackexchange.com/questions/2880738/is-there-any-conjecture-that-has-been-proved-to-be-solvable-provable-but-whose-d/2881223#2881223)).2018-08-15

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