Twin prime conjecture

There are infinitely many pairs of primes that differ by 2 — for example (11, 13), (17, 19), (29, 31). This has been believed since antiquity but never proven.

In 2013 Yitang Zhang made a breakthrough by showing that infinitely many prime pairs differ by at most 70 million; the Polymath project later reduced the bound to 246, but bringing it down to 2 remains open.