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Last Updated:  Friday, 4 April, 2003, 11:11 GMT 12:11 UK
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Prime number breakthrough
 By Dr David Whitehouse BBC News Online science editor

 Each one a prime
A pair of mathematicians has made a breakthrough in understanding so-called prime numbers, numbers that can only be divided by themselves and one.

Other mathematicians have described the advance as the most important in the field in decades.

It was made by Dan Goldston, of San Jose State University, and Cem Yildirim, of Bogazici University in Istanbul, Turkey. It has just been announced at a conference in Germany on Algorithmic Number Theory.

The advance is related to an idea called the twin prime conjecture. This idea, still unproved, is that there are an infinite number of pairs of prime numbers that differ only by two.

Number building

"Neither of us ever expected to get particularly good results by this method. It's actually completely amazing to me," says Goldston.

Commenting on the breakthrough, Hugh Montgomery, a mathematician at the University of Michigan in Ann Arbor, US, says that Goldston has really broken a barrier.

Primes have always fascinated mankind. The third century BC Greek mathematician Eratosthenes developed a way to find the prime numbers.

 Pierre Fermat was interested in primes
Over the years, mathematicians such as Pierre de Fermat in the 17th Century, Georg Riemann in the 19th Century and Godfrey Hardy in the 20th have advanced our understanding of these strange numbers.

One of the important things about primes is that they are the building blocks of the integers - whole numbers. Primes can be multiplied to obtain all of the other integers.

A curious observation is that primes occur in twins with a surprising regularity. For example: 11 and 13; 17 and 19; 29 and 31; 41 and 43; 59 and 61.

Just as with single primes, the frequency of twin primes decreases as one gets to larger numbers. But do they completely fizzle out beyond some very large number? That is the big question. Around a trillion, for instance, only about one in every 28 numbers is a prime.

'Incredible breakthrough'

To tackle this problem, Goldston did what clever mathematicians do when they want to solve a difficult problem - they avoid it. Or rather, he approached the dilemma by first tackling a more manageable piece of the problem.

He asked if it was possible to find prime numbers that might not be twins, but that were much closer together than average? After many years of study, he was able to show it was.

According to Brian Conrey, of the American Institute of Mathematics, the way Goldston went about solving the problem was just as important as the result.

"It's a brand new technique that opens the door," Conrey says. "A lot of the excitement is we don't know how far this thing is going to go. There are going to be a lot of applications, I think. It's an incredible breakthrough."

His paper is called Small Gaps Between Primes, and co-authored with Cem Yildirim. It places mathematicians closer to the tantalizing goal of identifying the frequency and location of twin primes.

"This result blows out of the water a whole line of previous records, as if someone were to run a three-minute mile," says Carl Pomerance, of Bell Laboratories, US.

The distribution of primes is closely related to one of the most renowned unsolved questions in mathematics, the Riemann hypothesis, which concerns an infinite sum of numbers called the zeta function.

In 2000, the Clay Mathematics Institute, US, offered \$1m to anyone who could settle the Riemann hypothesis. Goldston is optimistic that the new result will say something about the zeta function.

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