The twin prime conjecture

The statement. There are infinitely many pairs of primes that differ by 2.

In plain words. Primes thin out as numbers grow, but they never seem to stop bunching up in pairs. The largest known twin pair has 388,342 digits.

What you need first. Stage 10 (primes, sieves — Eratosthenes' sieve is the ancestor of the whole method), Stage 12 (the analysis that measures how primes are distributed).

The parts. (1) The heuristic count: about 1.32 · x/(ln x)² twin pairs below x, so infinitely many are expected. (2) Sieve theory: counting numbers that survive being divisible by small primes — powerful but limited by the “parity problem”. (3) Bounded gaps: Zhang (2013) proved infinitely many prime pairs differ by at most 70,000,000; the Polymath project and Maynard brought that to 246. (4) The remaining gap: from 246 down to 2 needs an idea nobody has.

What is known. Bounded gaps of 246 unconditionally, 6 assuming a strong conjecture; Brun's theorem says the sum of reciprocals of twin primes converges, so they are rarer than primes — but nobody knows if they run out.