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 "
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."