Find a close pair
Examples include 3 and 5, 11 and 13, and 29 and 31.
→Understand it in two minutes
Mathematicians now know that some fixed finite gap between consecutive primes occurs infinitely often. The best unconditional published bound is 246, but proving that the recurring gap can be exactly 2 remains open.
Prime numbers become less frequent, but they keep showing local clusters. Zhang proved that infinitely many consecutive prime pairs are separated by less than a fixed large number. Maynard and the Polymath project developed stronger methods and reduced explicit bounds. The remaining target is exact: replace a finite upper bound such as 246 with the specific value 2.
Watch primes become sparser
Average prime gaps grow, so an infinite supply of exact gap-2 pairs is far from automatic.
→Separate bounded gaps from twin gaps
A theorem guaranteeing infinitely many gaps no larger than 246 is major progress, but it does not identify gap 2.
Why people care
Fine prime structure
The conjecture probes correlations that are much subtler than the average distribution of primes.
Reusable sieve methods
The bounded-gap program created and sharpened techniques that apply to wider questions about prime constellations.
Open collaboration
The Polymath project demonstrated how distributed mathematical collaboration can improve explicit bounds.