The twin prime conjecture

Primes are the numbers divisible only by 1 and themselves: 2, 3, 5, 7, 11, and on forever. They are the atoms of arithmetic, since every other whole number is built by multiplying them together. And the closer you look, the stranger they get.

Twin primes are pairs that sit just two apart: 3 and 5, 11 and 13, 17 and 19. Far out among gigantic numbers, primes grow rarer and rarer, so you might expect these close pairs to dry up. The twin prime conjecture says they never do, that no matter how high you count, there is always another twin pair waiting somewhere ahead. It is one of the oldest unsolved problems in mathematics, simple to state and, so far, impossible to prove.

In 2013 it cracked open, from nowhere. A little-known lecturer named Yitang Zhang, who had spent years out of academia doing odd jobs, including a stretch making sandwiches at a Subway, proved the first real step: that pairs of primes keep coming close together forever, within some fixed gap, even if that gap was a hefty 70 million rather than 2. He was almost sixty and all but unknown. Within weeks he was famous, and within months other mathematicians had hammered his 70 million right down.

There are infinitely many primes; Euclid proved that over two thousand years ago. But they thin out as you climb. The prime number theorem makes this exact: near a large number \(n\), roughly one in every \(\ln n\) numbers is prime, so primes get sparser the higher you go.

That makes the gaps between consecutive primes grow on average, and you can even build runs of any length with no primes at all. So the real surprise is at the other end. Do primes ever stop coming close together? Twin primes, two apart, are as close as primes can get, aside from the single pair 2 and 3.

We know twins are rare. In 1919 Viggo Brun proved something striking: add up the reciprocals of all the twin primes and the sum settles on a finite value, about 1.9, now called Brun's constant. The reciprocals of all the primes, by contrast, add up to infinity. So twins really are scarcer than primes in general. What Brun's result could not say is whether there are finitely or infinitely many of them.

For a century no one could prove that primes keep coming close together at all, with any fixed gap. Then in April 2013 Yitang Zhang submitted a paper to the Annals of Mathematics proving there are infinitely many pairs of primes that differ by no more than 70 million. It was the first finite bound of its kind. The 70 million was enormous, and Zhang knew it, but the point was that the gap was finite and never closed completely. He had found the missing piece in a method by Goldston, Pintz and Yıldırım that had stalled just short of a finite bound.

His story was as unlikely as the result. After a PhD he drifted out of academia and worked a string of jobs, including at a Subway sandwich shop, before landing a lecturer post at the University of New Hampshire, where he worked in near-obscurity. He was about 58 when the paper appeared, an age by which most mathematicians are long past their best-known work.

Then the bound fell fast. A collaborative Polymath project crowdsourced improvements, and James Maynard, with Terence Tao, soon found a sharper method. Within roughly a year the gap was down from 70 million to 246. Under a stronger unproven assumption, the Elliott-Halberstam conjecture, the same machinery reaches a gap of 12, and a strengthened form gets to 6. But the final step, all the way to 2, the twin prime conjecture itself, is still out of reach.

How primes are distributed. The prime number theorem says the count of primes up to \(x\), written \(\pi(x)\), satisfies \(\pi(x) \sim x/\ln x\). A sharper estimate uses the logarithmic integral \(\mathrm{li}(x)\), and how tightly \(\pi(x)\) tracks \(\mathrm{li}(x)\) is governed by the Riemann Hypothesis, the Millennium Prize problem about the zeros of the zeta function. Prime gaps and the Riemann zeros are two faces of the same question.

How common should twins be? The Hardy-Littlewood conjecture predicts the density of twin primes precisely. The number of twin pairs up to \(x\) should be about \(2 C_2 \, x/(\ln x)^2\), where \(C_2 \approx 0.66\) is the twin prime constant. Every count we can actually do agrees with it closely, which is strong evidence it is true, but agreement is not proof.

Brun's sieve. Brun's 1919 theorem, that the reciprocals of twin primes converge to Brun's constant near 1.902, came from a genuinely new idea: a sieve that carefully over- and under-counts to bound how many twins there can be. Sieve methods became the central tool, but they ran into a famous wall, the parity problem, which on its own stops them from ever certifying that a number is prime rather than a product of two primes.

Zhang's breakthrough. The Goldston-Pintz-Yıldırım method of 2005 showed that good enough information about how primes spread across arithmetic progressions would force bounded gaps, but it needed a little more than the Bombieri-Vinogradov theorem could give. Zhang's achievement was to prove a workable version of that stronger distribution result for a restricted, well-behaved family of moduli, just enough to cross the line and pin the gap at 70 million.

Maynard and Tao. Months later James Maynard, and independently Terence Tao, replaced the one-dimensional sieve with a multidimensional one that was both simpler and far more powerful. It pushed the gap down to the low hundreds, eventually 246 through the Polymath collaboration, and proved more besides: that bounded-length intervals contain not just two primes but any fixed number of them, infinitely often.

Why 2 is so hard. Assuming the Elliott-Halberstam conjecture, which says primes are spread across arithmetic progressions about as evenly as anyone could hope, the methods reach a gap of 12, and a generalized form reaches 6. But the last stretch down to 2 runs straight into the parity problem, the deep limitation of sieves, and no one has a way around it. So the twin prime conjecture sits where it has for more than 150 years: almost certainly true, backed by overwhelming numerical evidence, now hemmed in to within a gap of 246, and still without a proof.