Twin prime pairs (p, p+2) shown in red. They appear to continue forever but the twin prime conjecture remains unproved.
In 2013 Zhang proved for the first time that infinitely many prime pairs are within a bounded distance. Within a year the bound fell from 70 million to 246.
Twin primes are prime pairs differing by 2: (3,5), (5,7), (11,13), (17,19), (29,31)... The twin prime conjecture that infinitely many exist is unproved. The twin prime constant C2 ≈ 0.66016 appears in the Hardy-Littlewood conjecture. In 2013, Zhang proved infinitely many prime pairs exist with gap less than 70 million. The Polymath project rapidly reduced this to 246. Whether the gap can be reduced to 2 (proving the conjecture) remains open.