## Game of proofs boosts prime pair result by millions

来源：未知 作者：令狐演 时间：2019-03-13 01:16:09

By Jacob Aron Sixty-five million sound like a big deal by any standard. That’s the size of the improvement that an online army of collaborating mathematicians has already made to a groundbreaking proof involving pairs of prime numbers, which was first announced just a few weeks ago. Though the improvement is big, mathematically speaking it amounts to a technicality. Still, the achievement showcases a new way of doing mathematics online. Since the proof appeared, mathematicians from across the world have been locked in an addictive race to tighten it up. The work relates to a longstanding problem called the twin prime conjecture. A prime number can only be divided by 1 and itself, and twin primes are those just two numbers apart, like 3 and 5, or 29 and 31. The conjecture, put forward in 1849, says there are an infinite number of these pairs, but no one has managed to prove or disprove it. Last month Yitang Zhang of the University of New Hampshire in Durham took an important step towards this goal by showing there are infinite number of primes that are separated by 70 million or less. It was the first time someone had put an upper limit on the gap between pairs of primes. Since then, mathematicians have been competing online to shrink the limit. Zhang proved that a set of 3.5 million numbers with a certain mathematical property produces an infinite number of similar sets and that these contain at least two primes. He then showed for a particular series of sets that these primes are at most 70 million apart. But with a fairly quick calculation it is possible to choose other sets with a smaller gap. Timothy Trudgian of the Australian National University in Canberra, Australia did just that last week, shrinking the prime gap to just under 60 million. Shortly after, Scott Morrison, also of the Australian National University, wrote a blog post in which he showed how to knock off another 40,000 or so. Other mathematicians then joined in, posting new records in the form of comments on Morrison’s blog all weekend. Morrison eventually reclaimed the crown with a gap of 13,008,612. Just today he further reduced it to 4,802,222 – shaving just over 65 million off Zhang’s initial proof. “This project is curiously addictive; I guess because it is much easier to make progress on it than with more normal research projects,” wrote Terence Tao of the University of California, Los Angeles, who also contributed to the calculations. This back-and-forth game of proofs has diminishing returns, as closing the gap becomes more difficult the smaller it gets. It is also not particularly important to mathematics, as Zhang’s key insight was proving that a maximum gap existed at all. “I think what this shows is that Zhang gets full marks for coolness in not trying too hard to optimize his bound,” wrote Timothy Gowers of the University of Oxford in an online discussion. It is, however, a shining example of a new way of doing mathematics collaboratively online which both Gowers and Tao are trying to promote via their Polymath project. Today Tao proposed that shrinking the prime gap and refining Zhang’s argument should become an official Polymath project. Any new results could be published under the pseudonym D. H. J. Polymath. Correction: Misleading references to “digits” that were in this story when it was originally published on 4 June 2013 have been removed. More on these topics: