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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$\begingroup$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:
$\endgroup$ 3 $\begingroup$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.
$\endgroup$ $\begingroup$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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