Introduction
By the blog author
Prime numbers (2,3,5,7,11,13,17,19,23,29…) are integers which can only be divided by themselves and by 1 to yield integral fractions of themselves. For a very long time, there has been speculation about the possible nature of primes. Some conjectures have been around for many generations – conjectures being notions that other mathematicians haven’t been able to prove nor to dismiss.
One of these conjectures is called the "twin primes conjecture," which speculates that there are an infinite number of prime numbers which are arithmetically only two numbers apart from each other. No one has been able to crack this conjecture to upgrade it to a proven mathematical truth or to eliminate it as ultimately untrue.
But a virtually unknown mathematician has come up with a proof that paves the road toward resolving the twin primes conjecture. He has shown 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."
It’s arcane. There’s a big difference between 2 and 70,000,000. But what has been proven amounts to very significant new mathematical knowledge. Here’s an excellent article on how this discovery occurred:
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Unheralded Mathematician
Bridges the Prime GapBy Erica Klareich, Simons Foundation, May 19, 2013
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.
By the blog author
Prime numbers (2,3,5,7,11,13,17,19,23,29…) are integers which can only be divided by themselves and by 1 to yield integral fractions of themselves. For a very long time, there has been speculation about the possible nature of primes. Some conjectures have been around for many generations – conjectures being notions that other mathematicians haven’t been able to prove nor to dismiss.
One of these conjectures is called the "twin primes conjecture," which speculates that there are an infinite number of prime numbers which are arithmetically only two numbers apart from each other. No one has been able to crack this conjecture to upgrade it to a proven mathematical truth or to eliminate it as ultimately untrue.
But a virtually unknown mathematician has come up with a proof that paves the road toward resolving the twin primes conjecture. He has shown 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."
It’s arcane. There’s a big difference between 2 and 70,000,000. But what has been proven amounts to very significant new mathematical knowledge. Here’s an excellent article on how this discovery occurred:
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Unheralded Mathematician
Bridges the Prime GapBy Erica Klareich, Simons Foundation, May 19, 2013
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."
….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.
http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/
This article was reprinted on Wired.com.
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