Mind the (Prime) Gap

Comeback kid. Goldston despaired of rescuing his proof, but a bright idea saved the day.

Despite early hiccups, number theorists say they have finally proved a key conjecture about prime numbers: that the smallest possible gap between two large prime numbers continues to shrink, relative to the natural logarithm of the smaller number, as the numbers increase.

Prime numbers--positive integers such as 2, 3, 5, 7, 11, and 13, which can't be broken down into smaller factors--become rarer as numbers get larger. On average, the gap between a large prime p and the next prime number is approximately the natural logarithm of p, written log p. Number theorists long ago proved that there is no upper limit on how large the gap can grow, relative to log p. But they did not know what happened to the size of the smallest possible gap as the numbers became larger.

Two years back number theorist Dan Goldston at San Jose State University in California and his colleague Cem Yildirim at Bogaziçi University in Istanbul, Turkey, announced a breakthrough--only to quickly learn that their proof contained a fatal error (Science, 4 April 2003, p. 32; 16 May 2003, p. 1066). But now, with the help of János Pintz of the Alfréd Rényi Mathematical Institute in Budapest, Hungary, they have unveiled a new proof, which confirms that the smallest possible gap continues to shrink relative to log p, as the numbers increase. Experts who have examined it say the proof is rock-solid--in part because it is much simpler than the earlier attempt. Mathematicians feel the basic result is not a surprise but it may help in tackling the long-unsolved "twin prime" conjecture. It holds that there are an infinite number of twin primes whose gap is 2. The list of such numbers starts with (3, 5), (5, 7), and (11, 13), and has been tabulated into the trillions. No one knows whether twin primes ever stop appearing but "The conjecture doesn't seem impossible to prove anymore," says Goldston, who gave a public presentation on the new proof at a number theory conference held from 18 to 21 May at the City University of New York.

"It's of enormous importance," says Brian Conrey, director of the American Institute of Mathematics in Palo Alto, California. "It's going to open the door to lots of stuff." Andrew Granville of the University of Montreal, Quebec, whose work helped torpedo the original flawed proof, agrees. "It's quite a turning point," he says.

Related sites
Number theory conference
Goldston's website
Info on Number Theory

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