Ivars Peterson's MathTrek

March 23, 1998

# A Prime Surprise

When it comes to prime numbers, expect the unexpected.

Last January, as part of a large international effort, Manfred Toplic in Austria identified a sequence of nine consecutive prime numbers, with each successive number 210 larger than its predecessor. That set the record for the largest known number of consecutive primes in arithmetic progression (see Nine Primes in a Row).

The achievement prompted Harvey Dubner in New Jersey, Tony Forbes in Great Britain, and Paul Zimmermann, Nik Lygeros, and Michel Mizony in France to launch a new campaign to find 10 consecutive primes in arithmetic progression. At that time, Dubner estimated that 500 participants in the search would require about 6 months to identify the required sequence, testing perhaps as many as 3 quadrillion numbers along the way.

Many of the 100 or so people who had helped with the nine-prime effort immediately signed up for the new quest and began checking numbers. To everyone's surprise, Manfred Toplic (the same!) set the new record, reporting on March 2 that he had found 10 consecutive primes in arithmetic progression. What incredible luck!

The record-holding sequence starts with the 92-digit number 100,996,972,469,714,247,637,786,655,587,969,840,329,509,324,689,190,041,803,603, 417,758,904,341,703,348,882,159,067,229,719, with each successive prime 210 larger.

Toplic, who works as a computer operator for an Austrian bank, is keenly interested in number theory and also participates in the Great Internet Mersenne Prime Search (see Another Prime Record).

Along the way, one of the 70 participants in the 10-prime search found a new set of nine consecutive primes, and others identified additional examples of eight consecutive primes in arithmetic progression.

Back in 1995, the record was only seven consecutive primes in arithmetic progression. Hundreds of examples have been found since, but these seven-prime progressions all involve numbers of more than 90 digits. In an entirely independent effort, Warut Roonguthai of Bangkok, Thailand, recently discovered a seven-prime sequence in which the first prime is only 58 digits long: 2,776,730,282,935,811,696,683,091,601,578,382,215,448,255,048,660,349,224,161.

"I will try to find a significantly smaller example," he says.

What about 11 consecutive primes in arithmetic progression? "We believe that a search for an arithmetic progression of 11 consecutive primes is far too difficult," Dubner and his colleagues say. "The minimum gap between the primes is 2,310 instead of 210, and the numbers involved in an optimal search would have hundreds of digits. We need a new idea or a trillionfold increase in computer speeds."

"So we expect the 10 primes record to stand for a long time to come," they conclude.