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Yitang Zhang: A prime-number proof and a world of persistence

报道出处: http://news.cnet.com/8301-17938_105-57618696-1/yitang-zhang-a-

prime-number-proof-and-a-world-of-persistence/

An exciting breakthrough by an academic little known before last year is

firing up mathematicians. Now even playwrights are getting in on the act.

Leslie Katz February 12, 2014 9:33 AM PST

This is a story about prime numbers and the man who took a giant step toward

solving a puzzle that has vexed mathematicians for centuries. That man, Yitang

"Tom" Zhang, was exiled to the Chinese countryside during Mao's Cultural

Revolution as a teen and forced to quit his studies and perform hard labor.

Later, after receiving advanced degrees in China and the US, he struggled to

find an academic position and at one point worked behind the counter at a

Subway sandwich shop.

"There's nothing wrong with working at a Subway, but normally these proofs,

these breakthroughs, are achieved by those that are working at Princeton,

Harvard, these kind of really elite places," Tony Padilla, a physics professor

at the UK's University of Nottingham, says in the Numberphile podcast below.

"And now we've got somebody who's literally come of nowhere, that no one

expected to produce this kind of results, and has done something impressive

that many great minds were unable to do."

Zhang himself says that "something impressive" probably has no practical

application, though prime numbers do extend beyond the realm of pure math into

more real-world uses.

"Fifty years ago no one would have dreamed that anything about primes had a

practical application, and large, secret primes are now the basis of some of

the cryptography that makes Internet commerce possible," David Eisenbud,

former head of the Mathematical Sciences Research Institute (MSRI) in Berkeley

, Calif., told Crave.

Those who follow math closely may have heard of Zhang last year when a paper

on his findings was accepted by the preeminent journal Annals of Mathematics

just a short three weeks after being submitted, and other mathematicians

responded to the research with exclamations like "beautiful," "stunned," and "

astounded."

Since then, the University of New Hampshire professor, who is in his late

fifties, has been invited to speak at institutions like Harvard, Princeton,

and UC Berkeley, and won two prestigious math prizes. His work has sparked a

frenzy among mathematicians the world over, who have collaborated through the

online Polymath project to further hone his proof.

Now, even artists are getting in on the story. San Francisco Bay Area

playwrights belonging to SF PlayGround, which develops new theatrical works,

recently attended a lecture on Zhang at the MSRI. The talk served as an

inspiration for original math-themed short plays, some of which will be

performed Monday night at the Berkeley Repertory Theater in Berkeley, Calif.,

at a program titled "A Passion for Primes."

Large, secret primes are now the basis of some of the cryptography that makes

Internet commerce possible.

--David Eisenbud, former head of the MSRI

Zhang's finding relates to the Twin Prime Conjecture, a number theory problem

that many attribute to the Greek mathematician Euclid. The conjecture holds

that there is an infinite number of prime numbers (numbers divisible only by 1

and themselves) that are only two numbers apart -- like 3 and 5 or 17 and 19.

These so-called twin primes occur often at the start of the lower end of the

number spectrum but become less frequent as numbers get higher.

Minding the gap

In his paper, titled "Bounded gaps between primes" and bearing his name alone

, Zhang attacked the problem by proving that the number of primes that are

less than 70 million units apart is infinite. While 70 million is a long, long

way away from 2, Zhang's work marked the first time anyone was able to assign

any specific proven number to the gaps between primes. (For a highly detailed

but clear explanation of Zhang's approach and results, read this Quanta

Magazine article by Erica Klarreich).

Zhang's work builds on a 2005 breakthrough by Daniel Goldston of California's

San Jose State University; Janos Pintz of the Alfred Renyi Institute of

Mathematics in Budapest, Hungary; and Cem Yildirim of Bogazici University in

Istanbul, who together showed that there will always be pairs of primes that

are much closer to each other than average spacing predicts.

They developed a "sieve" to filter for pairs of primes that are closer

together than average, as did Zhang. When his paper got accepted, Zhang said

he expected the formulas contained therein to form the basis for narrowing the

70 million gap. "We may reduce it," he said at the time.

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And they have. Terence Tao, a UCLA professor of mathematics and winner of the

esteemed Fields Medal, launched Polymath8 as a forum where mathematicians

could work to reduce that gap between 70 million and 2, which they did to 4,68

0 within a few months of Zhang submitting his paper.

That phase of the project was Polymath8a. In November, James Maynard, a

postdoctoral researcher at the University of Montreal, presented independent

work that built on Zhang's to further shrink the gap -- to 600. The second

phase of Polymath8, called Polymath8b, builds on Maynard's work.

"Given the substantial progress made so far, it looks like we are close to the

point where we should declare victory and write up the results," Tao wrote on

his blog Sunday.

"Right now the best bound on gaps between primes is 270," he told Crave, "

although we can get it down to the remarkably low level of 6 if we assume a

strong additional conjecture (the generalized Elliott-Halberstam conjecture)."

I didn't bring any notes, any books, any paper. And suddenly it came to me.

--Yitang Zhang, UNH math professor

Polymath 8 has been one of the most active, visible Polymath projects to date,

and Tao attributes the excitement surrounding it, in part, to Zhang's

compelling personal story -- after suffering the hardships of the Cultural

Revolution, Zhang earned a bachelor's and a master's degree from Peking

University, and completed a doctorate from Purdue, but had difficulty

obtaining a university job in the US. Tao also credits the relative simplicity

of the result, "which can be explained to any decent high-school math student

(unlike many recent advances in mathematics)."

In addition, he said, Zhang's proof came as something of a shock to the number

theory community. "The approach Zhang tried had been considered and discarded

by most other experts in the field."

Zhang himself, a self-described "shy person," said in a UNH statement that the

proof came to him during a vacation in Colorado, when he was feeling

particularly relaxed. "I didn't bring any notes, any books, any paper," he

said. "And suddenly it came to me."