Ivars Peterson's MathTrek

February 9, 1998

# Nine Primes in a Row

International teamwork has done it again.

Last November, researchers reported that they had identified a sequence of eight consecutive prime numbers, with each successive number 210 larger than its predecessor. That set a record for the largest known number of consecutive primes in arithmetic progression (see A Progression of Primes).

In general, an arithmetic progression consists of a set of integers a, a + d, a + 2d, a + 3d, and so on, where a is greater than 0 and d is greater than or equal to 2. In the record-setting case, d = 210 and a = 43,804,034,644,029,893,325,717,710,709,965,599,930,101,479,007,432,825,862,362,4 46,333,961,919,524,977,985,103,251,510,661.

With that result in hand, number aficionados Harvey Dubner of New Jersey, Paul Zimmermann of INRIA Lorraine in France, Tony Forbes of Great Britain, and Nik Lygeros and Michel Mizony of Claude Bernard University in Lyon, France, mounted a campaign and solicited computational help to find nine consecutive primes in arithmetic progression. They estimated that such an effort would require a total of about 200 computers running for a month.

On Jan. 15, Manfred Toplic of Klagenfurt, Austria, found the required sequence of primes. He was one of about 100 people, using 200 computers, who participated in the project by testing different ranges of trial values. Though it took twice as long as expected, the project was a great success, Dubner reports.

The record-holding sequence starts with the 92-digit prime number 99,679,432,066,701,086,484,490,653,695,853,561,638,982,364,080,991,618,395,774,0 48,585,529,071,475,461,114,799,677,694,651, with each successive prime 210 larger.

As a bonus, the project participants also found 27 new sets of eight consecutive primes in arithmetic progression and several hundred sets of seven primes.

"We are now considering how to proceed with the problem of finding ten consecutive primes in arithmetic progression," Dubner says. "It could take 500 people 6 months to find an answer."

"Eleven primes is another ball game entirely," he adds. "It would take at least [a trillion] times longer to solve than ten primes."

Says Forbes, "When we do find the ten primes, we expect the record to stand for a very long time to come."

References:

Dubner, Harvey, and Harry Nelson. 1997. Seven consecutive primes in arithmetic progression. Mathematics of Computation 66:1743-1749.

Guy, Richard K. 1994. Unsolved Problems in Number Theory. New York: Springer-Verlag.

Peterson, Ivars. 1995. Progressing to a set of consecutive primes. Science News 148(Sept. 9):167.

______. 1993. Dubner's primes. Science News 144(Nov. 20):331.

Ribenboim, Paulo. 1996. The New Book of Prime Number Records. New York: Springer-Verlag.