The scarcity of cluster primes.

The distribution of primes--whole numbers evenly divisible only by themselves and 1--has long intrigued mathematicians. More than 2,000 years ago, Euclid proved there are infinitely many primes. Nonetheless, primes gradually become scarcer as the numbers get larger.

To gain insights into the somewhat irregular distribution of prime numbers, mathematicians have studied a variety of subsets of all primes. The so-called twin primes, for example, consist of pairs of consecutive prime numbers that differ by 2, such as 41 and 43. Whether there are infinitely many twin primes remains one of the major unsolved questions in number theory.

Cluster primes define another puzzling subset. For a prime number p to qualify as a cluster prime, every even number less than p - 2 must be the difference of two primes, both of which are less than or equal to p. For example, 11 is a cluster prime because each of the even numbers 2, 4, 6, and 8 can be written as a difference between two of the primes 2, 3, 5, 7, or 11. The first 23 primes greater than 2 are all cluster primes. The smallest non-cluster prime is 97. In effect, the definition of a cluster prime encompasses a particular type of grouping among primes that mathematicians have found worthwhile investigating.

Mathematicians have observed that cluster primes become increasingly rare as primes get larger. New computations reveal that by the time the numbers reach 10 trillion, noncluster primes outnumber cluster primes by a ratio of about 325 to 1. In the January AMERICAN MATHEMATICAL MONTHLY, Richard Blecksmith and John L. Selfridge of Northern Illinois University in DeKalb and the late Paul Erdos report results suggesting that cluster primes are less numerous than twin primes. However, "we have no way of proving that either of these two collections is infinite," the mathematicians remark.