Ivars Peterson's
MathTrek |

February 9, 1998

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* + 2*d*, *a* + 3*d*, 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."

Copyright 1998 by Ivars Peterson

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.

Anyone interested in joining the search can obtain information at http://www.ltkz.demon.co.uk/ar2/10primes.htm. Progress reports are posted at http://www.desargues.univ-lyon1.fr/home/lygeros/prime10.html.

*(Source: http://www.maa.org/mathland/mathtrek_2_9_98.html)*

Comments are welcome. Please send messages to Ivars Peterson at mailto:ip@sciserv.org

Ivars Peterson is the mathematics and physics writer and online editor at
*Science News* (http://www.sciencenews.org/). He is the
author of *The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal
Defect*, and **The Jungles of Randomness: A Mathematical Safari*. His
current works in progress are an updated, 10th anniversary edition of *The
Mathematical Tourist* (to be published in 1998 by W.H. Freeman) and
*Fragments of Infinity: A Kaleidoscope of Mathematics and Art* (to be
published in 1999 by Wiley).

**NOW AVAILABLE The Jungles of Randomness: A Mathematical
Safari** by Ivars Peterson. New York: Wiley, 1997. ISBN
0-471-16449-6. $24.95 US.