Distribution of Primes [Hot off the press!]
---------- Forwarded message ---------- Date: Tue, 1 Apr 2003 23:03:54 -0800 (PST) Subject: FROM Science -- PRIMES * Prime Proof Shows How to Mind the Gaps * * * 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 Bogazii * 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."