The Year in Science: Mathematics

Sunday, January 08, 2006

Theorists Close In on Twin Prime Conjecture

In May, mathematicians Daniel Goldston of San Jose State University, Cem Yalçin Yildirim of Bogaziçi University in Istanbul and János Pintz of the Renyi Institute of Mathematics in Budapest reported they had discovered how prime numbers are dispersed among all numbers. Their announcement, backed by several world experts in the field of number theory, brought to an end two years of torment after a similar announcement in 2003 turned out to be wrong.

The new result is closely related to the famous twin prime conjecture, which says 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; 11 and 13 provide another example. The largest pair discovered so far (by computer) are numbers with 51,090 digits each.

Despite hundreds of years of effort, no one has proved the twin prime conjecture. The closest anyone has come was in 1966 when Chinese mathematician Chen Jingrun found that there are infinitely many primes p such that p + 2 is either prime or a product of two primes. (It is not known if this is true because the twin prime conjecture is correct or because products of two primes are allowed.)

One way to approach the twin prime problem is to look at the gaps between successive primes. For example, if p(1), p(2), p(3), . . . denotes the sequence of all primes, are there infinitely many values of n for which p(n + 1) – p(n) is less than 10, say, or less than 100? If you can find a gap for which there are infinitely many pairs of successive primes that differ by no more than that gap, maybe you can start to bring the gap down. If you get the gap down to 2, you will have proved the twin prime conjecture.

A variation is to forget the idea of looking for a fixed gap and instead compare the gaps between pairs of successive primes with the average size of p(n + 1) – p(n), and this is the approach that the team took. There is a single number D that measures this comparison precisely, and the method then comes down to trying to find the value of D. Others had tried this line of attack before and managed to achieve partial success. In 1940 the Hungarian mathematician Paul Erdös showed that D is less than 1; in 1966 two other mathematicians managed to show that D is at most 1¼2, subsequently bringing that down to 0.46650. In 1977 it was brought down even further, to 0.44254, and then to 0.2486 in 1986. Then along came Goldston, Yildirim, and Pintz. They didn't just improve the record; they smashed it once and for all, by showing that D is in fact 0.

That does not prove the twin prime conjecture, but it comes close. Of more significance is the fact that the new result tells us a lot more about how primes are distributed. To a mathematician, proving the twin prime conjecture would be like a geologist finding a new oil reserve. The Goldston-Yildirim-Pintz finding is like telling the geologist where all the world's major oil deposits are likely to be located.  —Keith Devlin 

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