How Number Theory Got the Best of the Pentium Chip

Chalk one up for number theory. With lurid accounts of the flaw in Intel's Pentium processor making front-page and network news, users of the personal computer chip in fields ranging from science to banking are finding cases where its faulty logic sends their computations awry. But the problem might have gone undetected for much longer if the chip had not slipped up months ago during a long series of calculations in number theory, raising the suspicions of a dogged mathematics professor. To other mathematicians, the discovery of the flaw by Thomas Nicely of Lynchburg College in Virginia emphasizes the value of number theory-the study of subtle properties of ordinary counting numbers-for providing quality control for new computer systems. By forcing a computer to perform simple operations repeatedly on many different numbers, number-theory calculations "push machines to their limits," says Peter Borwein of Simon Fraser University in Burnaby, British Columbia. Many computer makers have adopted these calculations as a shakedown test for systems intended for heavyduty scientific computation, and although the practice has yet to spread to personal computers, Borwein and some other mathematicians think that might be a good idea. Intel had actually found the flaw by other means after the chip had gone into production, but had decided that it was not likely to affect ordinary users. But the company hadn't counted on the use that Nicely had in mind. When he fired up a Pentium computer last March, Nicely was adding its number-crunching power to a project in computational number theory he had begun the year before. He was trying to improve on previous estimates of a number called Brun's sum, which is related to the distribution of prime numbers. The sequence of prime numbers-2, 3, 5, 7,11,13,17,19, etc.-isacontinuingsource of fascination to mathematicians. Since the time of Euclid, they have known that there are infinitely many primes, but although primes are relatively abundant early on, they become scarce among larger numbers. For example, roughly 23% of two-digit numbers are prime (21 of 90), but the figure for tendigit numbers is just 4%, and among hundred-digit numbers, the fraction of primes is less than half a percent. As a consequence, the gap between consecutive prime numbers tends to increase. However, every so often two odd numbers in a row turn out to be prime: 3 and 5, 41 and 43, 101 and 103, and 10,007 and 10,009, for example. Mathematicians conjecture that such "twin primes" pop up infinitely often. But in 1919, the Norwegian mathematician Viggo Brun proved that even if there are infinitely many twin primes, the sum obtained by adding their reciprocals-the sum (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... -converges to a finite value, much as the sum 1/2 + 1/4 + 1/8 + 1/16 + .... converges to 1. Brun's sum is known only to the first few digits, however-and even there, the accuracy is based on conjectures about the frequency with which twin primes occur. Number theorists think it's unlikely that clumps of twin primes