Twin primes are prime pairs that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)… They become rarer as numbers grow larger, but are they infinite? The twin prime conjecture that infinitely many exist: is one of the oldest unsolved problems in mathematics, posed by de Polignac in 1849.
The Hardy-Littlewood conjecture (1923) predicts that the number of twin prime pairs up to x is approximately 2C₂ · x / (ln x)². Here C₂ ≈ 0.66016 is the twin prime constant: a product over all odd primes p of p(p−2)/(p−1)².
In 2013, Yitang Zhang proved that there are infinitely many prime pairs with gap at most 70,000,000. The first finite bound ever proved. James Maynard independently reduced this to 600, and the Polymath project brought it to 246. The gap 2 (twin primes) remains out of reach.