Twin Prime Conjecture

Adding a correction proportional to $1/\ln p$ to a computation of Brun's Constant ending with $\ldots+1/p+1/(p+2)$ will give an estimate with error less than $c(\sqrt{p}\,\ln p)^{-1}$. An extended form of the conjecture states that

$$P_x(p,p+2)\sim 2\Pi_2\int_2^x {dx\over (\ln x)^2},$$

where $\Pi_2$ is the Twin Primes Constant. The twin prime conjecture is a special case of the more general Prime Patterns Conjecture corresponding to the set $S=\{0,2\}$.

See also Brun's Constant, Prime Arithmetic Progression, Prime Constellation, Prime Patterns Conjecture, Twin Primes