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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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    $\begingroup$ There is a very general conjecture in the theory of prime numbers -- Schinzel's Hypothesis H (en.wikipedia.org/wiki/Schinzel's_hypothesis_H) which covers your Q2. $\endgroup$ – Douglas S. Stones Dec 15 '10 at 1:05
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    $\begingroup$ @Douglas: Exactly the type of info I seek. Thanks! $\endgroup$ – Joseph O'Rourke Dec 15 '10 at 1:17
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    $\begingroup$ 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. $\endgroup$ – Mr Pie Feb 14 '18 at 13:08
  • $\begingroup$ @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. $\endgroup$ – Babelfish Aug 15 '18 at 16:20
  • $\begingroup$ @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 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" to $5,414$. I then saw on wikipedia that they proved there are infinitely many primes with a gap of $246$. $\endgroup$ – Mr Pie Aug 15 '18 at 21:31
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Sorry for the necropost, but I've only just now read this question. Both Q1 and Q3 are also considered in the first Hardy-Littlewood conjecture. In essence, the conjecture (also known as the k-Tuple conjecture) gives the density of such things as twin primes, cousin primes, and the like.

To be more precise, it considers pieces of the form $ p, p + a_1, p + a_1 + a_2, ..., p + a_1 + ... + a_k$ where infinitely many primes will be hit. So not only do pairs of the form (p, p + 2k) have their own conjecture, but so do triples such as (p, p + 2, p + 4) and such.

The actual conjecture can be found here: http://mathworld.wolfram.com/k-TupleConjecture.html

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    $\begingroup$ Please don't refer to it as the "first Hardy-Littlewood conjecture"; this is an oft-repeated mistake -- if anything, that refers to Goldbach's conjecture. The conjecture that p and p + 2k are simultaneously prime infinitely often with a specified density is their Conjecture B. The broader conjecture you mention is not in their famous 1923 paper (nor in any of the other joint Hardy-Littlewood papers I've read), though it does follow from the Schinzel-Sierpinski paper or that of Bateman & Horn. $\endgroup$ – Charles Apr 20 '11 at 13:02
  • $\begingroup$ @Charles: Let's just call a spade a spade $$$$ :) $\endgroup$ – The Chaz 2.0 Jun 19 '11 at 20:28
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    $\begingroup$ Triple primes $(p,p+2,p+4)$ do not have their own conjecture since there is just one example: $(3,5,7)$. $\endgroup$ – KCd Jan 8 at 20:32
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Modulo $30$: Twins: $11/13$, $17/19$, $29/31$. Cousins: $7/11$, $13/17$, $19/23$. Sexy: $1/7$, $7/13$, $11/17$, $13/19$, $17/23$, $23/29$. Each of the $8$ co-primes modulo $30$ are primes that beget an infinite number of primes, e.g., of form $7 + 30k$. If nothing untoward occurs in the netherlands, cousins and twins are equally dense, but half the density of sexy primes.

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

No, they are expected to have twice the density of the twin primes. This is because $(p,p+6)$ forms a different residue class than $(p,p+2)$ (and $(p,p+4)$). The Hardy-Littlewood $k$-tuple conjecture provides a way to estimate the amount of primes $p$ below a positive integer $x$ such that $p+6$ is also prime. If we denote this number by $\pi(x)_{(p,p+6)}$, we have:

$$ \pi(x)_{(p,p+6)} \sim 4 \prod_{p>=3} \frac{p(p-2)}{(p-1)^2} \int_2^x \frac{dt}{\log t^2}. $$

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

Yes.

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

Yes. In particular, the already mentioned Hardy-Littlewood conjecture provides a way to calculate an asymptotic density for such constellations, if indeed there are an infinite number.

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