I’m sure people will be working on it for years and then bring it down eventually.” What matters is that Zhang was able to show that the gap between adjacent primes cannot exceed a certain value. “The 70 million is not very important,” he says. Iwaniec is less concerned about that problem at the moment, though. “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. But Zhang stresses that this is an upper bound. Unfortunately for lonely primes, that distance is still quite large: 70 million. 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. Brainwaveīut in July last year, while at a friend’s vacation home, Zhang suddenly had a brainwave that let him make progress. However, there were small but significant obstacles to using the Goldston team’s method directly on the twin prime problem, Zhang said.

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. Zhang built on a 2005 paper by Daniel Goldston of San Jose State University in California and colleagues. “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. 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? “In number theory in particular, conjectures are pretty understandable,” says Henryk Iwaniec of Rutgers University in Piscataway, New Jersey.

Terence tao twin prime conjecture