张益唐证明了，存在无穷多组素数对(p,p+k)，使得k<=70,000,000。 虽然最终目标是证明存在无穷多组素数对(p,p+2)（著名的孪生素数猜想），但是这是重要的第一步，告诉人们孪生素数猜想是有希望攻克的。 值得一提是，张益唐年过50（？），到现在只是新罕布什尔大学的Lecturer，连faculty都不是。有评论说如果张益唐年轻点，菲尔兹奖不是没有希望的。 还有评论说张使用的是传统的解析数论的方法。张年轻时师从潘承彪，莫宗坚，专长是代数数论。后转向代数几何，没想到在知天命之年做出了解析数论的重大成果。 昨天，张益唐在哈佛大学给了报告，呈示他的证明。 下面是《自然》的报道。 -------------------------------------------------------------------------------------------------------------- First proof that infinitely many prime numbers come in pairs Mathematician claims breakthrough towards solving centuries-old problem. Maggie McKee 14 May 2013 Cambridge, Massachusetts It's a result only a mathematician could love. Researchers hoping to get '2' as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million. "That's only [a factor of] 35 million away" from the target, quips Dan Goldston, an analytic number theorist at San Jose State University in California who was not involved in the work. "Every step down is a step towards the ultimate answer." That goal is the proof to a conjecture concerning prime numbers. Those are the whole numbers that are divisible only by one and themselves. Primes abound among smaller numbers, but they become less and less frequent as one goes towards larger numbers. In fact, the gap between each prime and the next becomes larger and larger — on average. But exceptions exist: the so-called twin primes, pairs of prime numbers that differ in value by 2. Examples of known twin primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2195,000 - 1 and 2,003,663,613 × 2195,000 + 1. The twin prime conjecture says that there are an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. The problem has eluded all attempts to find a solution so far. A major milestone was reached in 2005 when Goldston and two colleagues showed that there are an infinite number of prime pairs that differ by no more than 161. But there was a catch. "They were assuming a conjecture which no one knows how to prove," says Dorian Goldfeld, a number theorist at Columbia University in New York. The new result, by Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever. The jump from 2 to 70 million is nothing compared to the jump from 70 million to infinity. "If this is right, I'm absolutely astounded," says Goldfeld. Zhang presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts, and the fact that the work did seem to use standard mathematical techniques led some to question whether Zhang could really have succeeded where others failed. But a referee report from the Annals of Mathematics, to which Zhang submitted his paper, suggests he did. "The main results are of the first rank," states the report, a copy of which Zhang provided to Nature. "The author has succeeded to prove a landmark theorem in the distribution of prime numbers. ... We are very happy to strongly recommend acceptance of the paper for publication in the Annals." Goldston, who was sent a copy of the paper, says he and the other researchers who have seen it "are feeling pretty good" about it. "Nothing is obviously wrong," he says. For his part, Zhang, who has been working on the paper since a key insight came to him during a visit to a friend's house last July, says he expects that the paper's mathematical machinery will allow for the value of 70 million to be pushed downwards. "We may reduce it," he says. Goldston does not think the value can be reduced all the way to 2 to prove the twin prime conjecture. But he says the very fact that there is a number at all is a huge breakthrough. "I was doubtful I would ever live to see this result," he says. Zhang will resubmit the paper, with a few minor tweaks, this week. |

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