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Topic: Solving a Riddle of (Twin) Primes
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Jerry P. Becker

Posts: 13,619
Registered: 12/3/04
Solving a Riddle of (Twin) Primes
Posted: May 21, 2013 7:16 PM
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From The New York Times [Science Times Section], Monday, May 20,
2013, p. D 8. See
http://www.nytimes.com/2013/05/21/science/solving-a-riddle-of-primes.html?_r=0
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Solving a Riddle of Primes

By Kenneth Chang

Three and five are prime numbers - that is, they are divisible only
by 1 and by themselves. So are 5 and 7. And 11 and 13. And for each
of these pairs of prime numbers, the difference is 2.

Mathematicians have long believed that there are an infinite number
of such pairs, called twin primes, meaning that there will always be
a larger pair than the largest one found. This supposition, the
so-called Twin Prime Conjecture, is not necessarily obvious. As
numbers get larger, prime numbers become sparser among vast expanses
of divisible numbers. Yet still - occasionally, rarely - two
consecutive odd numbers will both be prime, the conjecture asserts.

The proof has been elusive.

But last month, a paper from a little-known mathematician arrived
"out of the blue" at the journal Annals of Mathematics, said Peter
Sarnak, a professor of mathematics at Princeton University and the
Institute for Advanced Study and a former editor at the journal,
which plans to publish it. The paper, by Yitang Zhang of the
University of New Hampshire, does not prove that there are an
infinite number of twin primes, but it does show an infinite number
of prime pairs whose separation is less than a finite upper limit -
70 million, for now. (Dr. Zhang used 70 million in his proof -
basically an arbitrary large number where his equations work.)

"It's a deep insight," Dr. Sarnak said. "It's a deep result."

Dr. Zhang said he had been working on the Twin Prime Conjecture for
years and, like everyone else, failed. "I tried everything," he said.

Then, last July, "just very suddenly, an idea came to my mind," Dr.
Zhang said. "I was confident in this way I could prove it."

It took him another six months to fill in the details, but he appears
to be right. The paper has been accepted pending some small
revisions. "It's remarkable the speed this paper was dealt with," Dr.
Sarnak said.

Dr. Zhang's proof takes advantage of a 2005 paper by Daniel Goldston
of San Jose State University, Janos Pintz of the Alfred Renyi
Institute of Mathematics in Budapest and Cem Yildirim of Bogazici
University in Istanbul, which had shown there would always be pairs
of primes closer than the average distance between two primes. [See
http://arxiv.org/pdf/math/0508185v1.pdf ]

Still, in mathematics, closer does not necessarily mean two numbers
away, and experts were unable to make further progress on the
conjecture. "People tried, and after a few years, it seemed this was
really far away," Dr. Sarnak said.

Dr. Zhang also used techniques developed in the 1980s by Henryk
Iwaniec of Rutgers, Enrico Bombieri of the Institute for Advanced
Study and John B. Friedlander of the University of Toronto, adding
his own ingenuity to tie everything together in a way others had been
unable to.

"He got it," said Dr. Iwaniec, who has read Dr. Zhang's paper.
"There's no question about it."

The next step is reducing that 70 million separation, and Dr. Zhang
said "that should be very simple." But experts like Dr. Iwaniec said
bringing it all the way down to 2 - the full Twin Prime Conjecture -
would probably require more mathematical breakthroughs.

*********************************************
--
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244
E-mail: jbecker@siu.edu



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