Primes thin out forever, yet we can't prove whether pairs separated by just 2 ever stop appearing

The problem

Primes grow ever scarcer as numbers get bigger, yet twin pairs like 3 and 5, 17 and 19, keep turning up — the largest known pair has 388,342 digits. Whether they keep appearing forever is a question so natural the ancient Greeks could have asked it, formally posed by de Polignac in 1849, and still unanswered. In 2013 a virtually unknown lecturer, Yitang Zhang, stunned mathematics by proving that some gap below 70 million recurs infinitely often — the first finite bound in history. A frenzy of work slashed that bound to 246 within a year, but the final drop from 246 to 2 has been stuck ever since, blocked by a known theoretical obstruction called the parity problem.

Why it matters

The conjecture is the sharpest test of whether mathematics can control the fine-grained spacing of primes, not just their average density; the sieve techniques built for it now permeate number theory. Breaking the parity barrier that separates 246 from 2 would hand mathematicians a fundamentally new tool, with consequences far beyond twin primes.

Progress so far

• 1915Viggo Brun proves the sum of reciprocals of twin primes converges, founding modern sieve theory
• 2013Yitang Zhang proves infinitely many prime pairs differ by at most 70 million, the first finite bound in history
• 2014James Maynard, Terence Tao, and the Polymath project drive the bound down to 246, where it has stood since
• 2016the largest known twin prime pair, 2996863034895 × 2¹²⁹⁰⁰⁰⁰ ± 1 (388,342 digits), is discovered

References

1. Wikipedia — de Polignac 1849, Brun's theorem, the Zhang–Maynard–Polymath bounds, record twin primes
2. Quanta Magazine — Zhang's 2013 breakthrough
3. Quanta Magazine — the Polymath collaboration and Maynard's method shrinking the gap