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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 (en.wikipedia.org/wiki/Schinzel's_hypothesis_H) which covers your Q2. –  Douglas S. Stones Dec 15 '10 at 1:05
@Douglas: Exactly the type of info I seek. Thanks! –  Joseph O'Rourke Dec 15 '10 at 1:17

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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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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. –  Charles Apr 20 '11 at 13:02
@Charles: Let's just call a spade a spade $$$$ :) –  The Chaz 2.0 Jun 19 '11 at 20:28

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