# Progressing to a set of consecutive primes.

Searches for patterns among prime numbers have long served as stiff tests of the ingenuity and perseverance of mathematicians. In recent years, the use of computers in these prime pursuits has brought a steady stream of novel results.

Last week, Harvey Dubner, a semiretired electrical engineer in Westwood, N.J., and Harry L. Nelson, now retired from the Lawrence Livermore (Calif.) National Laboratory, added a new entry to the prime-number record book. They announced that they had found seven consecutive primes in arithmetic progression. The previous record had been six.

In other words, Dubner and Nelson unearthed a sequence of seven prime numbers--whole numbers exactly divisible only by themselves and one--in which each successive number is 210 larger than its predecessor, starting with the following 97-digit prime: 1, 089, 533, 431, 247, 059, 310, 875, 780, 378, 922, 957, 732, 908, 036, 492, 993, 138, 195, 385, 213, 105, 561, 742, 150, 447, 308, 967, 213, 141, 717, 486, 151.

Nelson had the idea of looking for such a string of consecutive primes after reading about these sequences in The Book of Prime Number Records (1989, Paulo Ribenboim, Springer-Verlag). Mathematicians and others had identified numerous examples involving six consecutive primes in arithmetic progression, but no examples of seven.

Nelson suggested the problem to Dubner, who had several personal computers specially modified and programmed to handle computations involving prime numbers (SN: 11/20/93, p.331). For mathematical reasons, they knew that the primes they were looking for had to be at least 210 apart, and they reasoned that numbers 90 to 100 digits long would be a good place to search for the required pattern. Initially, Dubner thought that the computations would take too long to be practical. But Nelson introduced a mathematical shortcut--a way of eliminating a large proportion of the candidate numbers--that considerably reduced the computation time needed for the search.

Dubner ended up using seven computers, running continuously for about 2 weeks, to find the sequence.

Now, Dubner and Nelson are thinking about taking the next step: going to eight consecutive primes in arithmetic progression. Dubner estimates that it would take about 20 times longer--at least 2.5 computer-years--to accomplish this search on his souped-up personal computers.

"But we can probably improve our method," Nelson says. "We need some advice from number theorists." By using better techniques and a larger number of computers, "there's a very real possibility that you could go to 10 consecutive primes," he adds.

Finding sequences of 11 or more such primes is vastly more difficult, however. Candidate numbers would have to be at least 1,000 digits long, Nelson estimates.

At the same time, there's probably no end in sight. Mathematicians have conjectured (but not yet proved) that in the infinite universe of whole numbers, there is no limit to the number of consecutive primes in arithmetic progression.