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.
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