## Proof that an infinite number of primes are paired

来源：未知 作者：郦刚煌 时间：2019-03-13 05:03:09

By Lisa Grossman Prime numbers just got less lonely. A proof announced this week claims to show that the number of primes with a near-neighbour that is also a prime number is infinite – although the “near-neighbour” primes may in fact be up to 70 million numbers away. The proof chips away at one of number theory’s most famously intractable problems, called the twin prime conjecture. A number is prime if you can’t divide it by anything but 1 and itself. Twin primes are primes that are only two numbers apart – like 3 and 5, 5 and 7, and 11 and 13. The largest known twin primes are 3,756,801,695,685 × 2666,669 + 1 and 3,756,801,695,685 × 2666,669 – 1, and were discovered in 2011. The twin prime conjecture states simply that there are an infinite number of these twin primes. Although simple in its concept, a proof of it has been stumping mathematicians since the idea was proposed in 1849 by French mathematician Alphonse de Polignac. “In number theory in particular, conjectures are pretty understandable,” says Henryk Iwaniec of Rutgers University in Piscataway, New Jersey. “But proving is another matter.” To make their work a little easier, mathematicians have aimed at answering a slightly different question: is there an infinite number of primes which have a neighbouring prime some finite distance away, even if that distance is much larger than 2? “My main result is just this: yes,” said Yitang Zhang of the University of New Hampshire in Durham at a seminar at Harvard University yesterday. Zhang built on a 2005 paper by Daniel Goldston of San Jose State University in California and colleagues. Typically, the gap between prime numbers grows for larger and larger numbers, but Goldston’s team showed that there always exist some primes that are very close together even in the realm of very large numbers. However, there were small but significant obstacles to using the Goldston team’s method directly on the twin prime problem, Zhang said. But in July last year, while at a friend’s vacation home, Zhang suddenly had a brainwave that let him make progress. He was able to exploit a technical detail to show that there is an infinite number of prime pairs that are separated by a measurable, finite distance. Unfortunately for lonely primes, that distance is still quite large: 70 million. But Zhang stresses that this is an upper bound. “These values are very rough,” he says. “I think to reduce them to less than one million or even smaller is very possible” – although mathematicians may need another breakthrough to reduce the distance all the way down to just 2 and finally prove the twin prime conjecture. Iwaniec is less concerned about that problem at the moment, though. “The 70 million is not very important,” he says. What matters is that Zhang was able to show that the gap between adjacent primes cannot exceed a certain value. “People will be stunned by the result. I’m sure people will be working on it for years and then bring it down eventually.” Iwaniec, who has made contributions to the twin prime problem but was not involved in the new work, has reviewed a paper presenting Zhang’s proof and cannot find an error in it. Zhang’s paper has been accepted for publication in the Annals of Mathematics. “His result is beautiful,” Iwaniec says. “He should enjoy his 15 minutes of fame.” In other prime number news, another mathematician has made progress on an equally intractable prime problem first posed by Christian Goldbach in 1742. Golbach suggested that every even number greater than 2 is the sum of two primes. Now Harald Helfgott of the École Normale Supérieure in Paris, France, has proved a related problem: the odd Goldbach conjecture, which states that every odd number above 5 is the sum of three primes. A proof of Goldbach’s conjecture would also prove the odd version, since you can then take an even number formed of two primes and add 3 to it to get an odd number formed of three primes. But Helfgott’s proof is unlikely to help mathematicians go in the other direction, says Terence Tao of the University of California, Los Angeles – so Goldbach’s original problem remains unsolved. Additional reporting by Jacob Aron More on these topics: