Twin, cousin, sexy, ... primes 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!

> **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.