Twin prime conjecture

Remarks

A pair of primes $(p, p + 2)$ is a pair of twin primes. The first pairs are $(3, 5)$, $(5, 7)$, $(11, 13)$, $(17, 19)$, $(29, 31)$. The conjecture is the case $k = 1$ of the conjecture of de Polignac: for each positive integer $k$, there are infinitely many primes $p$ such that $p + 2k$ is prime. The conjecture is open.

History

De Polignac stated the general conjecture in 1849. In 2013, Yitang Zhang proved that some gap $N < 7 \cdot 10^7$ occurs between infinitely many pairs of primes. Later work by Maynard and the Polymath project reduced the bound to $246$.