The Lecturer At Subway Who Solved A 100-Year-Old Prime Problem

Added: 2026-05-10

[00:00:00]3, 5, 7, 11, 13.

[00:00:05]The smallest primes.

[00:00:07]A pattern starts to peek out. 3 and 5 differ by 2.

[00:00:12]5 and 7 differ by 2. 11 and 13 differ by 2.

[00:00:16]These are called twin primes. Pairs of primes separated by exactly 2, the smallest possible separation since one of any two consecutive integers is even.

[00:00:28]The question is ancient. Look down the list, 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31. They keep appearing.

[00:00:40]As you go further, primes thin out. The chance two random integers near size n are both prime is about 1 over log squared [music] of n, vanishingly small.

[00:00:51]And yet twin pairs keep showing up.

[00:00:54]Computer searches have found over a quadrillion of them.

[00:00:58]Are there infinitely many?

[00:01:00]Euclid proved primes themselves are infinite around 300 BC. Twin primes, gap of 2, has stayed open for over 2,000 years.

[00:01:10]Brun showed the sum of reciprocals of twin primes converges, unlike primes themselves where it diverges. [music] So twins are genuinely rare. Hardy and Littlewood in 1923 conjectured the count of twins up to n grows like a constant times n over log squared n.

[00:01:28]Computers say this matches, but heuristics aren't proof.

[00:01:32]The conjecture has stared [music] mathematicians in the face for a century. Nobody had gotten near a proof.

[00:01:38]By 2010, the field's working assumption was this won't be solved anytime soon.

[00:01:45]Maybe never. Suppose proving infinitely many gaps of exactly [music] two is too hard. Mathematicians ask a softer question.

[00:01:54]Forget two.

[00:01:55]Are there infinitely many prime pairs separated by some bounded amount, any bounded amount? [music] Define a sequence. For each n, let pn be the [music] nth prime and consider the gap pn + 1 - pn.

[00:02:11]As n grows, the average gap grows like log of pn. Sometimes gaps are larger, sometimes smaller.

[00:02:19]The question, do small gaps keep happening forever or do they thin out [music] and disappear?

[00:02:25]In 2005, Goldston, Pintz, and Yildirim proved a striking result.

[00:02:31]The limit of the ratio of these gaps to log p, the smallest gap relative to the average, is zero.

[00:02:38]In words, prime gaps that are vanishingly small fractions of the average happen infinitely often.

[00:02:45]This was huge. It said primes don't behave like random numbers, where the smallest gaps would be on the order of log p.

[00:02:53]But it didn't say the gaps were bounded.

[00:02:56]The smallest gaps in their proof could grow just slower than log p. To get bounded, a fixed h such that infinitely many gaps are below h, they would have needed an additional ingredient.

[00:03:08]Stronger control over prime distribution in arithmetic progressions.

[00:03:13]They couldn't get it.

[00:03:14]They ended their paper noting the obstacle. The obstacle was a theorem of Bombieri and Vinogradov from 1965.

[00:03:22]It said primes are well distributed in arithmetic progressions modulo q on average over q up to roughly square root of x. To bound prime gaps, GPY needed it for q up to a higher power. The boundary, square root of x, was conjecturally not the truth. Elliott and Halberstam said you could push to x to 1 - epsilon.

[00:03:43]But nobody had proven anything beyond square root in general. Yitang Zhang was born in Shanghai in 1955.

[00:03:50]He learned mathematics from his father and from books his mother brought home.

[00:03:55]The Cultural Revolution interrupted his education. He was sent to a labor camp at 15 and worked there for 10 years. He didn't enter university until 1978 at age 23 when restrictions were lifted.

[00:04:09]He came to the United States in 1985, finished a PhD at Purdue in 1991.

[00:04:15]His thesis was on the Jacobian conjecture, a hard problem in algebraic geometry.

[00:04:21]The result didn't come together. He left Purdue without academic prospects.

[00:04:25]>> [music] >> For 7 years, he had no permanent job. He worked as an accountant.

[00:04:31]He worked at a Subway sandwich shop in Lexington, Kentucky. [music] Friends who knew his ability were quietly horrified.

[00:04:38]>> [music] >> In 1999, Zhang got a position as a lecturer at the University of New Hampshire.

[00:04:44]Lecturer is a title that means you teach, you don't get tenure, you don't get research support, you fade.

[00:04:51]He taught calculus. He had a small office, no graduate students, no funding. He worked on number theory in the cracks of his teaching schedule.

[00:05:00]He published a few papers, nobody noticed. [music] For 14 years, he kept thinking about prime gaps.

[00:05:07]The GPY work in 2005 had pointed at the obstacle. Zhang slowly in solitude attacked the obstacle. He didn't tell colleagues what he was working on.

[00:05:18]In April 2013, at age 58, Zhang submitted a 50-page paper to the Annals of Mathematics, the most selective journal in the field.

[00:05:28]Title: Bounded [music] Gaps Between Primes.

[00:05:32]The Annals reviewed it in 3 weeks.

[00:05:34]Usually, it takes a year. They accepted it on May 21st, [music] 2013. The theorem in Zhang's paper: There exists a positive integer H less than 70 million such that there are infinitely many pairs of primes pn + 1 and [music] pn satisfying pn + 1 - pn less than h. What does this say? It says prime gaps do not eventually grow without bound. In every interval extending to infinity, there is some bounded gap that keeps recurring forever.

[00:06:07]Some specific gap less than 70 million happens infinitely often. 70 million is enormous.

[00:06:14]The twin prime conjecture says you can take h equals two, a trillion times smaller than Zhang's bound.

[00:06:22]But twin primes is an existent statement about a specific number, two.

[00:06:27]Zhang's theorem is the first time anyone has proved the gap [music] stays bounded by anything. Bounded gaps had been a target for a century.

[00:06:35]Zhang ended its open status. The bound 70 million was an artifact of the proof, not a fundamental obstacle. The proof depended on a parameter that you could optimize.

[00:06:46]Zhang chose a value that made the analysis tractable, but not optimal. He knew the 70 million could be lowered. He just wanted to show the bound existed at all.

[00:06:57]The reaction was instantaneous. Andrew Granville reviewed the paper for the Annals and called it the most interesting thing he had seen in years.

[00:07:05]>> [music] >> Terrence Tao, who had been working on adjacent problems, posted within hours of the announcement. Henrik Iwaniec said, "This is the biggest result in analytic number theory this century."

[00:07:16]Zhang himself, contacted by reporters, said he was surprised by the attention.

[00:07:21]He had been working on the problem for over three years. He thought the result was correct, but he didn't expect anyone to care this much. Zhang's proof builds on the GPY method.

[00:07:33]Consider the sum over primes p up to x of the indicator that p + h is also prime, weighted by some carefully chosen function. By choosing the weights to peak when small clusters of integers are all prime, you turn the count of small gaps into a manageable sum.

[00:07:50]The sum splits into terms. The dominant terms involve sums of the von Mangoldt function over [music] arithmetic progressions. To bound them, you need control over how primes [music] distribute among residue classes modulo Q.

[00:08:04]The Bombieri-Vinogradov theorem gives this on average up to Q less than square root of X. GPY needed the same average distribution, but for Q up to a slightly higher power of X. Zhang's contribution was a refined version of Bombieri-Vinogradov restricted to Q with smooth factorizations.

[00:08:23]Q whose prime factors are all bounded by X to a small power. For these special moduli, >> [music] >> Zhang pushed the average distribution past square root of X into the territory needed. The technique was called Bombieri-Vinogradov with well-factorable weights.

[00:08:39]It was technical, hard, and clever.

[00:08:42]Zhang combined it with deep work on Kloosterman sums, exponential sums over residue classes, using estimates due to Deligne from arithmetic algebraic geometry. The bound on the Kloosterman sum, which Deligne proved as a corollary of his Weil conjectures work, gave Zhang the savings he needed.

[00:09:01]The actual chain of reasoning is intricate, but the structural picture is clear. Zhang took a tool that almost worked, the GPY sieve, and patched it with a sharper distribution result for primes over a restricted family of moduli.

[00:09:17]The patch closed the gap. Bounded gaps fell out. Within a week of Zhang's announcement, Terence Tao started Polymath 8, an online collaborative project on his blog inviting anyone to optimize the bound. [music] The structure of Zhang's proof was a long chain of estimates, each with a parameter. Each parameter could be tightened. The collective could explore the parameter space far faster than any individual.

[00:09:41]The bound dropped fast. Within days, 42 million. Within weeks, 10 million. By mid-July, less than 1 million.

[00:09:50]By the end of July, less than 400,000.

[00:09:53]By August, less than 5,000. Polymath 8 had a running tally on Tao's blog, updated in real time. Mathematicians around the world contributed improvements to specific lemmas, better numerical optimization, refined sieve constructions.

[00:10:10]By October 2013, the bound was around 4,000.

[00:10:15]The momentum slowed. The remaining factor, getting from thousands to [music] hundreds, would require new ideas, not just optimization.

[00:10:23]Then in November, James Maynard, a 26-year-old post-doc at Université de Montréal, posted a paper. Maynard had developed an independent proof of bounded gaps using a different sieve.

[00:10:36]His method, based on a multi-dimensional analog of GPY, gave a much smaller bound directly, 600.

[00:10:44]Maynard's proof was also conceptually cleaner.

[00:10:47]The sieve weights had a natural interpretation. The argument generalized to bounding the smallest k-tuples of primes, not just pairs, but triples, quadruples, and so on.

[00:10:58]With Maynard's machinery, Polymath could push further. By April 2014, a combined Polymath plus Maynard effort got the bound down to 246.

[00:11:09]That's where it sits today, 246, 23% above the bound from the strongest version of the Elliott-Halberstam conjecture, which would push it to 12.

[00:11:19][music] Maynard's contribution deserves its own beat. His sieve was different from GPY's.

[00:11:25]Where GPY used one-dimensional weights, functions of a single variable, Maynard used multi-dimensional weights, functions of K variables. The optimization problem for the right weight function became a constrained variational problem in K dimensions.

[00:11:41]The result was striking. The same general framework that gave Zhang his 70 million now gave Maynard 600, a 100,000-fold improvement with cleaner ingredients and no Kloosterman sum machinery.

[00:11:55]Maynard's work didn't use Deligne's deep theorem. It was, in technical terms, far more elementary.

[00:12:02]Maynard's method also gave more. It proved that for every K, there is a positive integer HK such that infinitely many sets of K primes [music] lie within an interval of length HK. Not just pairs, triples, quadruples, arbitrarily long bounded clusters. The bounded gap result [music] was the special case K equals 2.

[00:12:24]Maynard had shown how to prove a much stronger family of theorems with a much [music] simpler tool.

[00:12:29]He won the Fields Medal in 2022, in part for this work. The combined effect, Zhang's breakthrough plus Maynard's machinery plus Polymath's optimization, moved bounded gaps from a problem people couldn't approach to a problem with a concrete numerical bound.

[00:12:46]The twin prime conjecture is still open.

[00:12:49]But Zhang's paper proved that the problem is finite, that bounded gaps exist, and that finite barrier is what twin primes ultimately needs to overcome. The twin prime conjecture remains officially unsolved.

[00:13:03]246 is the best unconditional bound on prime gaps that recur infinitely often.

[00:13:09]Conditional on the generalized Elliott-Halberstam conjecture, the bound drops to 12. Conditional on a still stronger statement nobody knows how to prove, the bound is six.

[00:13:21]To reach two nucleuster, twin primes would need techniques nobody has invented yet. After Zhang's paper, the prizes came.

[00:13:30]The Cole Prize in 2014.

[00:13:32]The Ostrowski Prize in 2013.

[00:13:35]The Rolf Schock Prize in 2014. He went from a lecturer at New Hampshire to a full professor at the University of California, Santa Barbara.

[00:13:45]The lecture circuit found him.

[00:13:47]The world found him.

[00:13:49]Maynard finished his PhD shortly after publishing. He won the Fields Medal in 2022. The work he did at age 26 is now standard graduate level analytic number theory.

[00:14:01]Zhang himself, asked about the path to the proof, said this, "I was very calm.

[00:14:07]I am a slow worker.

[00:14:09]I had to work alone."

[00:14:12]The deeper lesson lives in the math. For a hundred years, mathematicians believed bounded gaps were essentially out of reach without the generalized Elliott-Halberstam conjecture.

[00:14:23]Zhang showed that wasn't true. A clever workaround, restricting to smooth moduli, exploiting Deligne's bound, gave the result without the unproven hypothesis.

[00:14:35]The barrier had a door. Yitang Zhang spent 14 years thinking about prime gaps in a small office at the University of New Hampshire while teaching calculus to undergraduates.

[00:14:46]Nobody asked him to do it. Nobody funded him.

[00:14:49]He had no graduate students, no peers in his subfield. He just kept working.

[00:14:54]Then, at age 58, he opened the door.