Are there infinitely many twin primes?
Pairs of primes just two apart, like 11 and 13, keep turning up however far we look. Nobody can prove they never stop.
The question
Prime numbers become rarer as numbers grow, but every so often two of them appear side by side, separated by just one even number: 3 and 5, 11 and 13, 17 and 19, 41 and 43. Such pairs are called twin primes. Computers have found them among numbers with hundreds of thousands of digits.
Do they go on forever? Euclid proved more than two thousand years ago that there are infinitely many primes. The same question for twin primes looks almost as simple. In 1849 the French mathematician Alphonse de Polignac stated a broader version: every even number appears infinitely often as the gap between consecutive primes. For the gap of 2, nobody has been able to prove it.
What we know
- Twin primes thin out, but slowly. In 1919 Viggo Brun showed that the sum of the reciprocals of all twin primes converges to a finite number, unlike the sum for all primes. So twin primes are much rarer than primes. But his result doesn't say whether they stop.
- In 1923 G. H. Hardy and J. E. Littlewood predicted precisely how many twin primes there should be up to any given size. Counts by computer agree with their formula to remarkable accuracy, far beyond where any counterexample would be plausible.
- In 1973 Chen Jingrun proved that there are infinitely many primes p such that p + 2 is either a prime or a product of two primes. This is as close as classical methods came.
- In 2005 Daniel Goldston, János Pintz and Cem Yıldırım showed that primes sometimes come much closer together than their average spacing. Their method narrowly failed to give a fixed bound.
- In 2013 Yitang Zhang, then a little-known lecturer at the University of New Hampshire, proved that there are infinitely many pairs of primes less than 70 million apart. It was the first proof that the gaps don't grow without limit.
- Within months a public online collaboration, the Polymath project, reduced the bound. James Maynard, and independently Terence Tao, found a simpler and more powerful method, and by 2014 the bound stood at 246. Maynard received the Fields Medal in 2022, partly for this work.
Main hypotheses
The conjecture is true
Almost all mathematicians believe there are infinitely many twin primes, and that the Hardy–Littlewood formula describes them correctly. The numerical evidence and the heuristic reasoning both point the same way. The question is not really whether, but how to prove it.
Improving the sieve
Most of the progress has come from "sieve methods", which filter out numbers with small factors. Assuming a widely believed conjecture about how evenly primes are distributed, the Elliott–Halberstam conjecture, the bound drops to 12, and under a stronger version to 6.
A new idea is needed
There is a known obstacle. Sieve methods cannot distinguish numbers with an odd number of prime factors from those with an even number — the so-called parity problem, identified by Atle Selberg. This seems to block every sieve method from reaching a gap of exactly 2. Most experts think a proof will need a genuinely new idea.
Where understanding ends
We know that prime gaps don't grow forever: some pairs of primes stay within 246 of each other, infinitely often. We don't know whether the bound can be pushed all the way down to 2, and the parity problem suggests that current methods can't do it.
The deeper puzzle is the nature of the primes themselves. They are completely determined by simple arithmetic, yet locally they behave as if scattered at random. Probabilistic models predict their behaviour, including twin primes, astonishingly well. But no one knows how to turn this apparent randomness into proofs. The same gap between what we can predict and what we can prove separates us from the Riemann hypothesis.
What would settle it
- A method that breaks the parity barrier of sieve theory — or a fundamentally different approach to prime gaps.
- A proof of the Elliott–Halberstam conjecture, which would at least bring the bound down to 12.
- A counterexample is not realistic: there is no finite search that could show that twin primes stop, so the question can only be settled by proof.
Sources
- de Polignac, A. (1849). Six propositions arithmologiques déduites du crible d'Ératosthène. Nouvelles annales de mathématiques (1re série), 8, 423–429. (Scan: Numdam.)
- Brun, V. (1919). La série 1/5 + 1/7 + 1/11 + 1/13 + … où les dénominateurs sont nombres premiers jumeaux est convergente ou finie. Bulletin des Sciences Mathématiques (2e série), 43, 100–104, 124–128. (Scan of the original volume: Internet Archive.)
- Hardy, G. H., & Littlewood, J. E. (1923). Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes. Acta Mathematica, 44, 1–70.
- Chen, J. R. (1973). On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Scientia Sinica, 16, 157–176.
- Goldston, D. A., Pintz, J., & Yıldırım, C. Y. (2009). Primes in tuples I. Annals of Mathematics, 170(2), 819–862.
- Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics, 179(3), 1121–1174.
- Polymath, D. H. J. (2014). Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences, 1(1), 12.
- Maynard, J. (2015). Small gaps between primes. Annals of Mathematics, 181(1), 383–413.