Wednesday, April 02, 2003

On the Distribution of Prime Numbers It looks like a major breakthrough has been made on a classic mathematical problem. Prime numbers are those divisible only by one and themselves. Thus, the numbers 2, 3, 5, 7, 11, 13, 17 and 19 are all primes. For over two centuries mathematicians have been studying the distribution of prime numbers among the integers. For example, there are four primes smaller than ten, but only 26 primes less than 100. Up to a trillion, on average one in twenty-eight numbers is prime.

However, though the appearance of individual primes is impossible to predict, their average behavior is quite regular. The prime number theorem tells us that for large values of x, the number of primes smaller than x is very close to x divided by log x. However, this does not imply that you can't find pairs of primes that are much closer together than the prime number theorem would lead you to believe. For example, if p and q are two large, consecutive primes, with p larger than q, a naive application of the prime number theorem might lead you to believe that p-q divided by log q is pretty close to one. Actually, it has been known for some time that you can find infinitely many pairs of primes p, q for which that quotient is smaller than 1/4.

The newly announced result, proven by D. Goldston, and C. Yildirim, is that actually for any fraction, no matter how small, you can find infinitely many pairs of consecutive primes p, q so that p-q divided by log q is smaller than that fraction.

The result is presently going through peer review, but hopefully it will hold up. If it does it represents a major leap forward in our understanding of the distribution of prime numbers. A brief explanation of the result can be found in this article from Science News. A more detailed summary is available from the American Institute of Mathematics here.

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