A prime number is one whose only positive divisors are 1 and itself. The early primes — 2, 3, 5, 7, 11, 13, 17 — are familiar to schoolchildren. As you move further out along the number line, primes become noticeably rarer. The prime number theorem tells us that the average gap between consecutive primes near a number N grows roughly like the natural logarithm of N. By the time you reach numbers with a few hundred digits, primes are typically separated by gaps in the hundreds.

A twin prime is a pair of primes that differ by exactly two — for example (11, 13), (17, 19), (29, 31), and so on. They are easy to spot at small numbers. They keep appearing at much larger ones. The largest pair known, discovered in 2016, has 388,342 digits.

The Twin Prime Conjecture, attributed to the French mathematician Alphonse de Polignac in 1849, claims there are infinitely many such pairs. By informal reasoning, this is surprising. As primes become sparser on average, the chance of two primes happening to land next to each other ought to drop toward zero. The conjecture says it does not.

For more than 150 years, the conjecture resisted every attack. The breakthrough came in April 2013, when the mathematician Yitang Zhang, then at the University of New Hampshire, proved that there are infinitely many pairs of primes whose gap is less than 70 million. The result was the first finite, unconditional bound on prime gaps. Within months, an online collaborative project organized by Terence Tao — Polymath 8 — and new methods from James Maynard drove the bound rapidly downward. By April 2014, it had been reduced to 246, where it has stayed.

To prove the original conjecture, the bound has to reach 2. As of 2026, no one has gotten there.

A pattern any patient child can spot remains, in the strict mathematical sense, unproven.