We're 99.9% sure this pattern is true, but no one can prove it

In April 2013, the Annals of Mathematics received a 50-page proof from an unknown person who had previously worked at a subway restaurant. The proof addressed the twin prime conjecture, which states there are infinitely many prime pairs separated by just one number (e.g., 11 and 13). Experts initially expected to find a mistake quickly but were surprised to find the proof held up, leading to a breakthrough.

As numbers increase, primes become rarer, with the average gap between consecutive primes growing roughly as the natural logarithm of N. For example, around 100 the average gap is about 4.6, and around 1000 it is 6.9. Despite this trend, twin primes are still found at large numbers, such as 1,000,037 and 1,000,039 after a million, and a record pair with 388,342 digits each. This empirical evidence suggests twin primes persist, but a proof requires more than checking finite examples.

In 1923, Hardy and Littlewood developed a method to estimate the number of twin primes using the prime number theorem, which gives the probability a large number near N is prime as 1/ln(N). For a pair (N, N+2), the naive probability is about 1/(ln(N))^2, but they added a correction factor for prime non-independence. The resulting estimate matches actual twin prime counts extremely closely—for example, up to 1 trillion, the error is only 0.001%—but it remains a heuristic, not a rigorous proof.

Despite the heuristic's accuracy, it cannot guarantee twin primes never stop; as Terry Tao noted, there could be a 'conspiracy' where primes forbid their neighbors from being prime. A rigorous mathematical proof is needed, but it is extremely difficult. One early attempt was by Norwegian mathematician Viggo Brun during World War I, working in isolation in Norway, aiming to prove the twin prime conjecture but ultimately making limited progress.

Brun realized that sieving by all primes up to the square root of X causes error terms to accumulate uncontrollably. By weakening the sieve—only sieving up to X^(1/10)—he gained control over the error term. The trade-off was that survivors could have up to nine prime factors, so he proved infinitely many pairs two apart where each number has at most nine prime factors. This technique was later refined to seven, then three, and finally by Chen Jingrun in 1973 to primes P where P+2 has at most two prime factors, the closest approximation to the twin prime conjecture without proving it.

A different approach asks: what is the smallest gap between two consecutive primes? Instead of allowing one number to be composite, both must be prime, but the distance between them is reduced. On average, consecutive primes are about the natural logarithm of N apart. By 1988, mathematicians had proven that primes sometimes come within roughly a quarter of that average gap. In 2005, Goldston, Pintz, and Yildirim (GPY) shocked the community by proving the gap could be made an arbitrarily small fraction of the average—0% of the average—meaning primes get infinitely often as close as you like, but still not a fixed bounded number.

In 2005, the American Institute of Mathematics gathered the world's top experts—including GPY, Andrew Granville, and Kannan Soundararajan—for a week in California with the explicit goal of proving a bounded gap between primes. A young graduate student attended and later recalled that Soundararajan showed it was impossible with the existing tools. The consensus was that the method had hit a wall, and the student abandoned the problem for his thesis. However, Yitang Zhang was not at this meeting.

Yitang Zhang grew up in China and moved to the U.S. around age 30 for a PhD in mathematics, but never got recommendation letters and struggled to find a job. He lived in his car for a time and worked odd jobs for seven years, including keeping books and sorting receipts at Subway. In his spare time, he would drive to the local library to read number theory books and journals. In 1999, a friend helped him get a lecturer position at the University of New Hampshire, allowing him to focus on math full-time. Zhang recalled imagining as a child that he would one day solve a major math problem.

After exhausting work with no results, Yitang Zhang visited a friend in Colorado in summer 2012. While waiting for a concert, he stepped into the backyard alone, hoping to see deer, but none came. As he walked and thought, the key idea suddenly struck him: instead of counting primes in many arithmetic progressions with all sorts of step sizes (as GPY had done), he focused on a special class of step sizes built only from small prime factors. This allowed him to reorganize error terms so most canceled out, pushing past the half barrier by a tiny fraction — just 1 over 584.

Zhang submitted his proof to the Annals of Mathematics on April 17, 2013. The journal receives claims of the Riemann Hypothesis daily, so they expected to find a mistake quickly. However, as expert referees flipped through the paper, they kept finding that each potential obstacle was elegantly handled — like laying a carpet that fits perfectly into every corner. By the end of the week, they had reconstructed the proof and confirmed everything was correct. Zhang's stencil had 3.5 million slots across a span of 70 million, and proving two slots always catch primes yielded a bounded gap of 70 million.

After Zhang's result, Terence Tao led the Polymath online collaboration to optimize the method. The upper bound dropped rapidly — month by month, week by week, even day by day — from 70 million down to 4,680. Meanwhile, young postdoc James Maynard, working independently with Andrew Granville in Montreal, developed a completely orthogonal approach. His advisor explicitly warned him not to work on it full time, expecting failure, but Maynard persisted. Within months, he brought the gap down to 600 and proved something more: his method could get three primes in a bounded window, and it had nothing to do with the exponent one half.

The long-assumed fundamental limit of one half turned out to be a pure mirage. GPY's average was stuck at two times theta, but Maynard's average grew roughly like theta over two times the natural logarithm of K (where K is the number of slots in the stencil). All Maynard needed was enough slots — any number greater than zero, not 0.50101. Curiously, Terence Tao independently arrived at the same approach and told Ben Green about it. This revealed that the half barrier was never a true mathematical limit, just an artifact of previous methods.