PALO ALTO, CALIFORNIA--In an unexpected breakthrough, two mathematicians have brought the erratic behavior of enormous prime numbers into dramatically sharper focus. Last week, speaking at the American Institute of Mathematics (AIM) here before an audience of 50 number theorists who had been buzzing about the new result all week, Dan Goldston of San Jose State University, California, described how he and Cem Yildirim of Bogaziçi University in Istanbul, Turkey, had proven that primes get more and more "clumpy" as they get larger.
The distribution of primes--integers that can be divided evenly only by themselves and 1--has vexed mathematicians for centuries. Primes may pop up in clumps, such as the numbers 101, 103, 107, 109, and 113, or at huge intervals. One of number theory's most celebrated open questions, the Twin Prime Conjecture, states that "twin primes"--those that crop up two numbers apart, such as 17 and 19--keep appearing forever as numbers get bigger. But like much else about prime gaps, its truth or falsehood remains a mystery. "[Primes] grow like weeds among the natural numbers, seeming to obey no other law than that of chance," number theorist Don Zagier wrote in 1977.
Goldston and Yildirim's proof takes a huge step toward understanding how "weedy" the primes are. Earlier mathematicians had shown that primes get sparser as they get larger. If n is a prime number, then the gap to the next prime will, on average, be the natural logarithm of n, or log n. But no one knew how clumpy the spacing is. Can two consecutive primes fit into a much smaller gap than log n? And can many primes fit into a single log-n interval?
The new proof answers both of the above questions in a single stroke. It shows that the shortest gaps between primes continue to shrink relative to the average gap (although, unfortunately for twin-prime aficionados, they could still be much larger than 2). What's more, there is no upper limit to the number of primes that can squeeze into the space "allotted" for one.
"This is the biggest excitement that prime number theory has seen since 1965," says Hugh Montgomery of the University of Michigan, Ann Arbor. That's when Enrico Bombieri showed that there's an infinite number of gaps of less than half the average size. Bombieri, now at the Institute for Advanced Studies in Princeton, New Jersey, agrees that Goldston and Yildirim have produced a "magnificent proof."
Goldston's site 
The Prime Pages 
A calculator for factoring numbers and identifying primes