More than 2,000 years ago, Euclid of Alexandria (325–265 B.C.) provided a simple proof that the sequence of prime numbers continues forever. A prime is a whole number (other than 1) that’s evenly divisible only by itself and 1. This definition leads to the following sequence of numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, and so on.

Suppose there is a finite number of primes, Euclid argued. This means there’s also a largest prime, *n*. Multiply all the primes together, then add 1: (2 x 3 x . . . x *n*) + 1. The new number is certainly bigger than the largest prime.

If the initial assumption is correct, the new number can’t be a prime. Otherwise, it would be the largest. Hence, it must be a composite number and divisible by a smaller number. However, because of the way the number was constructed, any known prime, when divided into the new number, leaves a remainder of 1. Therefore, the initial assumption can’t be correct, and there can be no largest prime.

Primes often occur as pairs of consecutive odd integers: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. These so-called twin primes are scattered throughout the list of all prime numbers. However, there’s no proof yet that there are infinitely many pairs of primes that differ by only 2.

There’s now hope that the matter will finally be resolved.

The distribution of primes follows a remarkably simple pattern: The average spacing between primes near a number *x* is the natural logarithm of *x*, a number closely related to the number of digits in *x*. At the same time, the gap between consecutive primes can often be much smaller or much larger than average. Indeed, mathematicians have proved that there are infinitely many pairs of primes that are closer than one-quarter of the average spacing.

More than a year ago, Daniel A. Goldston of San Jose State University and Cem Y. Yildirim of Bogaziçi University in Istanbul announced a proof that, given any fraction, no matter how small, there are infinitely many prime pairs closer together than this fraction of the average. In other words, tight clusters of primes show up among the whole numbers no matter how large the numbers are.

However, Andrew Granville of the University of Montreal and Kannan Soundararajan of the University of Michigan soon found a flaw in the purported proof. So it was back to the drawing board.

Now, with the help of Janos Pintz of the Rényi Mathematical Institute of the Hungarian Academy of Sciences in Budapest, Goldston and Yildirim have circumvented the flawed approach and completed the proof, confirming that the spacing between consecutive primes is sometimes very much smaller than the average spacing.

The new proof is much shorter and expressed in terms more familiar to number theorists than was the previous effort. In technical language, the theorem states that, for any positive number *x*, there exist primes *p* and *p’* such that the difference between *p* and *p’* is smaller than *x* log *p*.

The proof of an even stronger result about the size of these gaps is now in circulation among mathematicians, but it hasn’t yet been fully verified.

At a presentation at the American Institute of Mathematics in Palo Alto, Calif., on May 24, Goldston suggested that, based on results to date, a proof of the twin prime conjecture may not be far away.