Twin Prime Conjecture

The twin prime conjecture asserts that there are infinitely many primes $p$ for which $p+2$ is also prime. As catalogued here it is the gap-$2$ case of de Polignac's 1849 conjecture that every even number occurs infinitely often as a difference of consecutive primes [Polignac1849]; the full all-even-gaps assertion is a strictly stronger relative and is not duplicated in this record.

The decisive modern progress concerns bounded gaps. Zhang proved that infinitely many pairs of distinct primes differ by at most a fixed absolute constant [Zhang2014BoundedGaps], and Maynard's multidimensional sieve method [Maynard2015SmallGaps], refined collaboratively by the Polymath project [Polymath2014Gaps], reduced the admissible bound dramatically. These theorems yield infinitely many prime pairs at some bounded distance, but the sieve machinery cannot at present force any particular even gap, let alone the gap $2$.

The conjecture remains open: what is missing is an argument that pins the recurring bounded gap down to exactly $2$.