Text: Posted on Wed, Mar. 26, 2003 story:PUB_DESC SJSU math professor breaks barrier HE AND TURKISH COLLEAGUE SOLVE NUMBER THEORY By Glennda Chui Mercury News Dan Goldston, a math professor at SJSU, has solved an important problem in number theory relating to prime numbers. More photos (RICHARD KOCI HERNANDEZ / MERCURY NEWS) Dan Goldston, a math professor at SJSU, has solved an important problem in number theory relating to prime numbers. Working with a Turkish colleague, a San Jose State University math professor has solved one of the most important problems in prime number theory -- a solution that took him 20 years. ``There wasn't any rush, you know,'' Dan Goldston said. ``You just work away. And really, neither of us ever expected to get particularly good results by this method. It's actually completely amazing to me.'' Mathematicians described the advance -- announced at a conference in Germany -- as the most important breakthrough in the field in decades. Like many mathematical developments it has no immediate practical application but may open the door to a wealth of further advances. ``He's done something that a lot of people thought couldn't be done, really,'' said Hugh Montgomery, a mathematician at the University of Michigan in Ann Arbor. ``He's really broken a barrier.'' Goldston became fascinated years ago with something called the twin prime conjecture. This is the idea -- still unproved -- that there are an infinite number of pairs of prime numbers that differ only by two. Primes are numbers that can be divided only by themselves or by 1 without leaving a remainder. The smallest twin primes are 3 and 5. The largest? Well, people come up with bigger ones all the time. The biggest ones as of Tuesday were, in mathematical notation, 33,218,925 times two to the 169,690th, minus 1, and its closest prime neighbor. ``That's about a 17,000-digit number,'' Goldston said. ``They're not hard to find.'' At any given moment, in fact, the top 20 twin primes can be found on the Web at www.utm.edu/research/ primes/lists/top20/ twin.html. The question is, do these twin primes keep occurring indefinitely, up into the realm of zillions and bazillions? Or do they just fizzle out at some point? Most mathematicians suspect they keep going, off into infinity. But proving that has been infinitely difficult. So Goldston picked off a more manageable piece of the problem: Can you always find prime numbers that may not be twins, but that are much closer together than average? Taking into account, of course, the fact that the bigger numbers get, the sparser primes become. Working with Cem Yalcin Yildirim of Bogazici University in Istanbul, Goldston was finally able to say yes. ``It's hard to know where this is going to lead,'' Montgomery said. ``It could be that this is going to open up a path to twin primes or something like that.'' Brian Conrey directs the American Institute of Mathematics in Palo Alto, where Goldston will give a talk on his work on Friday. He says the way Goldston and Yildirim went about solving the problem is just as important as the result. ``It's a brand-new technique that opens the door,'' Conrey said. ``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.'' Goldston seems a bit surprised by it all. Colleagues describe him as funny, in a low-key kind of way -- not the kind of guy who pushes himself forward. But they add that he has been amazingly prolific, writing 33 mathematical papers over the years despite a full-time teaching load. A hint of his sense of humor can be found on his Web site, which features a photo of Goldston, seemingly dozing off, as two small kids climb on his back. He and his wife, Ryoko, have three children -- Shota, 7, Aiko, 5, and Makoto, 3. The couple met when she took one of his classes at San Jose State. ``He's so funny. He's always joking around. His lectures are hilarious,'' Ryoko said. ``He's sort of working on this while we watch TV,'' she said. ``He doesn't even have a study. His study is the dining room table. I think he's always thinking about math in his head, but he doesn't show it too much. ``He doesn't even own a suit. Maybe after this we're going to have to get him a suit or two.'' IF YOU'RE INTERESTED Dan Goldston's Web site is at www.math.sjsu.edu/~goldston/ An account of his work is at http://aimath.org/.

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