Unheralded Mathematician Bridges the Prime Gap

by Erica Klarreich

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1991 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

“Basically, no one knows him,” said Andrew Granville, a number theorist at the Université de Montréal. “Now, suddenly, he has proved one of the great results in the history of number theory.”

Mathematicians at Harvard University hastily arranged for Zhang to present his work to a packed audience there on May 13. As details of his work have emerged, it has become clear that Zhang achieved his result not via a radically new approach to the problem, but by applying existing methods with great perseverance.

“The big experts in the field had already tried to make this approach work,” Granville said. “He’s not a known expert, but he succeeded where all the experts had failed.”

**The Problem of Pairs**

Prime numbers — those that have no factors other than 1 and themselves — are the atoms of arithmetic and have fascinated mathematicians since the time of Euclid, who proved more than 2,000 years ago that there are infinitely many of them.

Because prime numbers are fundamentally connected with multiplication, understanding their additive properties can be tricky. Some of the oldest unsolved problems in mathematics concern basic questions about primes and addition, such as the twin primes conjecture, which proposes that there are infinitely many pairs of primes that differ by only 2, and the Goldbach conjecture, which proposes that every even number is the sum of two primes. (By an astonishing coincidence, a weaker version of this latter question was settled in a paper posted online by Harald Helfgott of École Normale Supérieure in Paris while Zhang was delivering his Harvard lecture.)

Prime numbers are abundant at the beginning of the number line, but they grow much sparser among large numbers. Of the first 10 numbers, for example, 40 percent are prime — 2, 3, 5 and 7 — but among 10-digit numbers, only about 4 percent are prime. For over a century, mathematicians have understood how the primes taper off on average: Among large numbers, the expected gap between prime numbers is approximately 2.3 times the number of digits; so, for example, among 100-digit numbers, the expected gap between primes is about 230.

But that’s just on average. Primes are often much closer together than the average predicts, or much farther apart. In particular, “twin” primes often crop up — pairs such as 3 and 5, or 11 and 13, that differ by only 2. And while such pairs get rarer among larger numbers, twin primes never seem to disappear completely (the largest pair discovered so far is 3,756,801,695,685 x 2

^{666,669}– 1 and 3,756,801,695,685 x 2

^{666,669}+ 1).

For hundreds of years, mathematicians have speculated that there are infinitely many twin prime pairs. In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.

Since that time, the intrinsic appeal of these conjectures has given them the status of a mathematical holy grail, even though they have no known applications. But despite many efforts at proving them, mathematicians weren’t able to rule out the possibility that the gaps between primes grow and grow, eventually exceeding any particular bound.

Now Zhang has broken through this barrier. His paper shows that there is some number

*N*smaller than 70 million such that there are infinitely many pairs of primes that differ by

*N*. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.

The result is “astounding,” said Daniel Goldston, a number theorist at San Jose State University. “It’s one of those problems you weren’t sure people would ever be able to solve.”

**A Prime Sieve**

The seeds of Zhang’s result lie in a paper from eight years ago that number theorists refer to as GPY, after its three authors — Goldston, János Pintz of the Alfréd Rényi Institute of Mathematics in Budapest, and Cem Yıldırım of Boğaziçi University in Istanbul. That paper came tantalizingly close but was ultimately unable to prove that there are infinitely many pairs of primes with some finite gap.

Instead, it showed that there will always be pairs of primes much closer together than the average spacing predicts. More precisely, GPY showed that for any fraction you choose, no matter how tiny, there will always be a pair of primes closer together than that fraction of the average gap, if you go out far enough along the number line. But the researchers couldn’t prove that the gaps between these prime pairs are always less than some particular finite number.

GPY uses a method called “sieving” to filter out pairs of primes that are closer together than average. Sieves have long been used in the study of prime numbers, starting with the 2,000-year-old Sieve of Eratosthenes, a technique for finding prime numbers.

To use the Sieve of Eratosthenes to find, say, all the primes up to 100, start with the number two, and cross out any higher number on the list that is divisible by two. Next move on to three, and cross out all the numbers divisible by three. Four is already crossed out, so you move on to five, and cross out all the numbers divisible by five, and so on. The numbers that survive this crossing-out process are the primes.

The level of distribution is known to be at least ½. This is exactly the right value to prove the GPY result, but it falls just short of proving that there are always pairs of primes with a bounded gap. The sieve in GPY could establish that result, the researchers showed, but only if the level of distribution of the primes could be shown to be more than ½. Any amount more would be enough.

The theorem in GPY “would appear to be within a hair’s breadth of obtaining this result,” the researchers wrote.

But the more researchers tried to overcome this obstacle, the thicker the hair seemed to become. During the late 1980s, three researchers — Enrico Bombieri, a Fields medalist at the Institute for Advanced Study in Princeton, John Friedlander of the University of Toronto, and Henryk Iwaniec of Rutgers University — had developed a way to tweak the definition of the level of distribution to bring the value of this adjusted parameter up to 4/7. After the GPY paper was circulated in 2005, researchers worked feverishly to incorporate this tweaked level of distribution into GPY’s sieving framework, but to no avail.

“The big experts in the area tried and failed,” Granville said. “I personally didn’t think anyone was going to be able to do it any time soon.”

**Closing the Gap**

Meanwhile, Zhang was working in solitude to try to bridge the gap between the GPY result and the bounded prime gaps conjecture. A Chinese immigrant who received his doctorate from Purdue University, he had always been interested in number theory, even though it wasn’t the subject of his dissertation. During the difficult years in which he was unable to get an academic job, he continued to follow developments in the field.

“There are a lot of chances in your career, but the important thing is to keep thinking,” he said.

Zhang read the GPY paper, and in particular the sentence referring to the hair’s breadth between GPY and bounded prime gaps. “That sentence impressed me so much,” he said.

Without communicating with the field’s experts, Zhang started thinking about the problem. After three years, however, he had made no progress. “I was so tired,” he said.

To take a break, Zhang visited a friend in Colorado last summer. There, on July 3, during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said.

Zhang’s idea was to use not the GPY sieve but a modified version of it, in which the sieve filters not by every number, but only by numbers that have no large prime factors.

“His sieve doesn’t do as good a job because you’re not using everything you can sieve with,” Goldston said. “But it turns out that while it’s a little less effective, it gives him the flexibility that allows the argument to work.”

While the new sieve allowed Zhang to prove that there are infinitely many prime pairs closer together than 70 million, it is unlikely that his methods can be pushed as far as the twin primes conjecture, Goldston said. Even with the strongest possible assumptions about the value of the level of distribution, he said, the best result likely to emerge from the GPY method would be that there are infinitely many prime pairs that differ by 16 or less.

But Granville said that mathematicians shouldn’t prematurely rule out the possibility of reaching the twin primes conjecture by these methods.

“This work is a game changer, and sometimes after a new proof, what had previously appeared to be much harder turns out to be just a tiny extension,” he said. “For now, we need to study the paper and see what’s what.”

It took Zhang several months to work through all the details, but the resulting paper is a model of clear exposition, Granville said. “He nailed down every detail so no one will doubt him. There’s no waffling.”

Once Zhang received the referee report, events unfolded with dizzying speed. Invitations to speak on his work poured in. “I think people are pretty thrilled that someone out of nowhere did this,” Granville said.

For Zhang, who calls himself shy, the glare of the spotlight has been somewhat uncomfortable. “I said, ‘Why is this so quick?’” he said. “It was confusing, sometimes.”

Zhang was not shy, though, during his Harvard talk, which attendees praised for its clarity. “When I’m giving a talk and concentrating on the math, I forget my shyness,” he said.

Zhang said he feels no resentment about the relative obscurity of his career thus far. “My mind is very peaceful. I don’t care so much about the money, or the honor,” he said. “I like to be very quiet and keep working by myself.”

Meanwhile, Zhang has already started work on his next project, which he declined to describe. “Hopefully it will be a good result,” he said.

Nassim Nicholas Taleb

ZHANG'S BARBELL, OR WHY ONE SHOULD BE AN ACADEMIC LOSER.

Yitang Zhang, more than the brilliant and profoundly no-bullshit mathematicial geniuses (such as Groethendieck or Pereleman), should be a role model for researchers (the exception to my via negativa that focuses solely on inverse role models). While other researchers worked on their "publications", salary increases, and "H-ratio", he quietly worked on his problem, getting deeper and deeper, barbelling it with lower employment (such as working for a sandwich store or as a math instructor). 22 years after his doctorate, he only had 19 academic citations. He wasn't a lone genius, rather he had the perfect profile of the academic loser.

He vindicates the argument in Antifragile that the success of the country rector in British Science during the 19th Century wasn't because academia wasn't as developed as it is today. It is simply because academia has never been the best place for scholarship.

Lavinia Teodorescu very inspiring!

16 hours ago · Like

Neenyi Ayirebi-Acquah Similar to Einstein who spent ten years at the Swiss Patent office whilst developing his theory of relativity. Great insight NNT.

16 hours ago · Like · 4

Pietro Bonavita I'm guessing the problem is that the system whereby a scientist is judged by his publications has become like a sport, and as such it can be "gamed". The original olympic disciplines were based on what a iron-age warrior would have to perform in his military duties: they had relevance to the real world, and the sportsman was also a practicioner. I think that this kind of validation should be reintroduced in all "sport-like" system, starting of course from sport itself, but including also "competitive academia": get it back to practical application, as much as you can, and measure THAT.

16 hours ago · Like · 14

Claudio Virili The advantage of being out of the mainstream, and use you own ideas to reach the goal (+ perseverance and self-confidence).

16 hours ago · Like · 8

Elia El Khazen Marwan El Khazen

16 hours ago · Like

Liu Zhi Guan Respect.

16 hours ago · Like

Blake Frank Too many compromises and too much noise in academia. The only advantage is access to library and people with specialist knowledge.

16 hours ago · Like · 5

David Boxenhorn Nassim, this is a real-life Tatar Steppe story. It is an illustration of the most distressing (to me) part of The Black Swan (and one reason why I love the book). I'm not at all sure its message is that people like Zhang should be role models. On the contrary, it is a warning that this path is perhaps heroic, but it will surely lead to contempt and ridicule - and only possibly to success.

16 hours ago · Like · 19

Sheeja Panicker some how reminds me of the movie ' a beautifull mind'' and the book by sylvia nasar

16 hours ago · Edited · Like · 2

Slava Fazebook I wonder how he was as a sandwich store employee (his mind obviously being constantly occupied with math). Probably a lousy one:)

15 hours ago · Edited · Like · 4

Marius de Villiers Must be the time he worked at subway, when the idea came to him. Subway should produce a "prime sandwich"

16 hours ago via mobile · Like · 7

Cemil Şinasi Türün Super example of a Black Swan event. Even Bombieri and Freedlander could not find this result from their own method. This obscure researcher did not know it was so, he kept looking at the 'wrong' places. Until finding the road. Thank you for sharing.

16 hours ago · Like · 3

Marc Fleury I appreciate the argument, that academe may not be the only place for scholarship. I do serious study outside of academe (20 years after my PhD) and not having to do the 'rat race' is a privilege that was common place in the 18,19 century. Being a gen...See More

15 hours ago · Like · 7

Andrew Yang any survivorship bias here? dont know how many people were like zhang and failed.

15 hours ago via mobile · Like · 11

Adam Sim Joyce Shi Sim

15 hours ago · Like

Ben Brady This also underscores the point of application-blindness in innovation - in that it was an ancient method applied in a new (less optimised) manner which yielded the breakthrough .. Bravo Zhang

15 hours ago · Like · 1

Nassim Nicholas Taleb Andrew Yang but we have a large datasets of academic contributions and it doesnt look impressive compared to the resources put into the business. Zhang is more like the historical norm.

14 hours ago · Like · 16

Vergil Den I wonder about his process: weeks of thinking about it at odd times then fits of inspiration, burning the mid-night oil. He probably had some semblance of a life too.

14 hours ago via mobile · Like · 7

Sanjeev Solanki Amazing!

14 hours ago via mobile · Like

Jonathon Richter A bit broad to compartmentalize everything scholars do under "academia" and act as if it is all homogenous. Take a look under the hood across the disciplines at various levels of the Carnegie class institutions and you'll see a lot of different species of research.

14 hours ago via mobile · Like · 1

Moin Rahman A corporate sinecure, that pays high to think little -- and uses its PhDs for providing commentary on the issues of the day or serve as quality control inspectors, with almost no original contributions, is another example of LOSERS, which a good number of old-school, Big Corporations excel at churning out these days.

13 hours ago · Edited · Like · 4

Pedro Zonari Grothendieck had skin, soul and bones in the games. He despised academia (after only 20 years as a researcher). Ok, he went a little nuts in the early nineties, but overall it's someone who has worked alone most of his time (even as a student)

13 hours ago via mobile · Like · 3

Ollie Frolovs "Experts in his field", "prominent mathematical journals" - so much pompousness "in his field", it seems.

13 hours ago via mobile · Like

Nassim Nicholas Taleb Pedro Zonari but Grothendieck was identified as a genius by every breathing person around him. An intolerant genius.

13 hours ago · Like · 6

Mark Hubey There are extremes. At the other extreme are 90+ percent of students who are taught not to work on problems that will take longer than 10 minutes. Most researchers slowly built up this duration up to a month, several months, or several years. He is one of those that worked on the problem over decades. Most scientists will keep working on such problems over a life time (that is mathematical scientists usually).

13 hours ago · Like · 1

Didier Clement And he is in his 50s when it is said that math is a young man game....

12 hours ago · Like · 1

Nathan C. Perry "Zhang said he feels no resentment about the relative obscurity of his career thus far. 'My mind is very peaceful. I don’t care so much about the money, or the honor,' he said. 'I like to be very quiet and keep working by myself.'"

12 hours ago · Like · 2

Russell Foltz-Smith Tis a beautiful thing. Taken in whole and without reductive analysis of his way or of academia, it's just beautiful to see someone take on a problem and chase it all the way down.

12 hours ago via mobile · Like · 2

Dru Stevenson These posts make me feel better about how few citations I have...

12 hours ago · Like · 4

Andy Catsimanes A great example of the difference between pursuing goods of excellence internal to a practice (that are impossible to achieve without virtue), vs. external goods (e.g. money, tenure, notoriety).

12 hours ago · Edited · Like · 1

Alessandro Sergi I always remember my judo teacher: To build one champ you need 100 tourists ... Let us not be too hard with academia ... There is also a place for non-geniuses in this world. Academia is an institution with good sides and bad sides. Honestly, notwithstanding everything, I do not see a better gym for the mind. Not all academics must be geniuses. I believe they should be the bones of the body of knowledge in a society (and also teachers ... would you really want to be taught by a lonely genius?). Geniuses will bring the flesh and the blood to the body of knowledge. I believe the trouble is in the openness (or lack of) of the academia. In my opinion, this the crux of the problem. There should be ways for academia to recognize lonely geniuses without prejudice or fights to defend little useless closed "gardens" ...

12 hours ago · Like · 4

Jeffery LeMieux I'm impressed that the gentleman is in his 50's. Rare for a mathematician to make breakthroughs of this magnitude later in life. But there are many kinds of intelligence, and perseverance is underrated today. IQ is a measure of intelligence based on a ratio between intellectual age and physical age. If a person keeps studying and learning as they age, won't they accumulate over time and possess more than a higher IQ young person who lacks experience or time to develop? In some fields, possibly. Intellectual and artistic turtles (myself included,) keep working on your passion!

11 hours ago · Like · 3

Baz Dafnidis Certainly a role model at that! Thank you for this Nassim!

11 hours ago · Like · 2

Jean-Christophe Alario Selection is also an important part of our education system that you can't deny. 19th century had no process of selection and was obviously missing many potential talents. That said, I agree one one thing the chase for publications, reputation and industry money is worst way to lead fundamental research like that one.

11 hours ago · Like · 1

Nassim Nicholas Taleb I know from long long conversations with Mandelbrot that the "young man's game" it is not factual. Mandelbrot was in his fifties when he started fractals. He said that he witness the reverse empirical evidence. Those who boomed early faltered quickly. He often used the case of Whitehead who did his real work when he was in his 70s.

11 hours ago · Edited · Like · 21

Jonathon Richter so... a couple exceptions means that the rule is a poor one?

11 hours ago · Like

Andy Chang Awesome story!

10 hours ago via mobile · Like

Alexander Boland Forget the "genius" part, I'm impressed that Zhang somehow had the guts and the endurance to deal with what must have been endless condescending sneers from people who thought he should "get ahead." How does one build that kind of social endurance without resorting to total hermitage?

10 hours ago · Edited · Like · 7

Jean-Christophe Alario I am not talking about the "early genius" (they are the outliers), I'm just saying that Mandelbrot would have never managed to achieve his research without learning the basis at Ecole Polytechnique when he was 20. And this was done thanks to a selectio...See More

10 hours ago · Like

Jean-Christophe Alario @Jonathon Richter actually the exception are the early geniuses like Evariste Galois or Mozart. The majority of researchers reach their pinnacle after 40, 50 or even more. If at 20 you're better to learn by heart 200 pages of theorems and proves, learning how to search need experience and wisdom.

10 hours ago · Like · 1

Robert Smits Not a nice story at all. A talented student goes to Purdue, has an egomaniac advisor who puts him to work on his pet project which demands more resources than a grad student has. 7 years of exploitation later he is without a job. He clings to his pa...See More

10 hours ago via mobile · Like · 1

Nassim Nicholas Taleb Jean-Christophe Alario this is not true about Mandelbrot. He was incapable of doing anything except his own way, no derivations, just visualization. He wasted his time at ecole Polytechnique and suffered -he never fit there. His method was intuition and it took him a while to revert to his style of visual conjectures without proofs.

10 hours ago · Like · 10

David Boxenhorn young man's game : math :: beginner's luck : gambling addiction

9 hours ago · Like · 3

Walter Marsh I'm not a mathematician, but this young man's game idea strikes me as real BS, an academic party line that I read as "we have to get them young so they'll be part of the club". I've heard this in my area since day one, and it also feels like a club member/bully's way of reinforcing their position. Thank you for the post, Nassim. It's very encouraging!

9 hours ago via mobile · Like · 5

Jason Skonieczny How do you think this phenomenon shakes out in the humanities and social sciences?

8 hours ago · Like

Vergil Den This Academic Loser and the like in other disciplines seem to be both heretic and artist in one. I wonder if the measure of the AF of a society (or some other construct) can be measured by the number of these losers (up to a point)?

8 hours ago via mobile · Like · 3

Jeffery LeMieux In the visual arts it's a mixed bag. The idea of the youthful artistic genius is always repeated, mostly on the basis of revolutionary work. Byron, Shelly, were young poets, Picasso young when first "discovered" by Stein, but their artwork is very emo...See More

8 hours ago · Like · 2

Walter Marsh he had the "aha" in a flaneur-like moment - before leaving for a concert.

7 hours ago · Like · 2

Aaron Elliott I think this shows how academia, and 9-5 employment in general, suffer from suppression of volatility - by demanding and rewarding constant (small) output, you never have the chance to go deep into an unproductive period to do something really big. Th...See More

6 hours ago · Like · 11

Ann Marsh def agree with final statement.

4 hours ago · Like

Paul Wehage I'm behind this one 1000 percent!

about an hour ago · Edited · Like · 1

Phillip F. Crenshaw Sublime.

29 minutes ago · Like

Alexander Boland Also, I feel somewhat reassured. I've been chasing an invention/idea for years, and have felt nervous about the mediocre trajectory of my life on the surface. Too much exposure to celebrities, ack!

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