Westfield State College senior mathematics majors Jeffrey P. Vanasse and Michael E. Guenette, working under the direction of Mathematics Department faculty members Marcus Jaiclin and Julian F. Fleron, have made a significant new discovery in the mathematical field of number theory. They have discovered the first known example of a 3 by 3 by 3 generalized arithmetic progression (GAP).

Most easily thought of as a 3 by 3 by 3 cube (similar to a Rubik’s cube puzzle) made up of 27 primes, their discovery begins with 929 as its smallest prime ends with 27917 as its largest prime. The intervening 25 primes are constructed by adding combinations of the numbers 2904, 3150, and 7440 in an appropriately structured method.

“Such an object was known to exist and its approximate size had been loosely estimated,” Fleron said. “However, a blind search would require checking more cases than can be feasibly checked by all existing modern computers each running for the next million years. Instead, the group used knowledge of the structural relationships between the potential candidates to greatly reduce the potential candidates to be checked.”

An algorithm to check the necessary cases – still easily hundreds of trillions of cases – was programmed using a Linux version of the computer language C++.

“This breakthrough is another indication that our Mathematics Department is working on a world-class level,” said Evan S. Dobelle, president of Westfield State College. “The college is very proud of these students and their professors for taking the initiative to practice cutting-edge mathematics.”

“We were worried that it might take months to run based on our estimates,” Guenette said. “Yet initial tests showed the algorithm running at a hopeful speed.”

“We were always optimistic, but the first tests got us really excited that our method would be successful,” Vanasse said.

The team broke the search up into groups of cases for each of the researchers to run on separate computers. Within days the first known example of a 3 by 3 by 3 GAP was found – one with largest prime of 197,957.

Having succeeded in finding the first known example, and now having a strict bound on the size of the largest prime, the group set to work finding other 3 by 3 by 3 GAPs – in particular, the smallest such. They were successful, showing there are exactly three 3 by 3 by 3 GAPs of primes with largest prime less than 50,000, the smallest example being that described above.

The students and faculty members are hopeful that their work will aid number theorists who continue to work on elusive patterns that lurk within the mystery of the prime numbers.

With estimates for the largest prime in a 4 by 4 by 4 or 3 by 3 by 3 by 3 GAP being near 5 quadrillion (that’s a 5 followed by 15 zeroes) the group is fairly certain that their record for finding the highest dimensional GAP will stand for quite some time, Fleron said.

Guenette of Easthampton, Mass., and Vanasse of Chicopee, Mass., both plan on attending graduate school in mathematics following graduation. Their work on this problem has provided them with a valuable introduction to the efforts of a working research mathematician.

For Jaiclin and Fleron, and the rest of the Westfield State Mathematics Department, it is another opportunity to share with and involve undergraduates what they love best: doing significant mathematics and advancing the frontier of human knowledge.

Fleron said the team’s work was inspired by the recent discoveries of the young Australian mathematician Terence Chi-Shen Tao, now a professor at the University of California, Los Angeles. “Many prominent number theorists are working simply to understand the implications of these discoveries,” he said. “Now Westfield State College students are playing a role, as well.”

British mathematician Andrew Granville, now on the faculty of the Université de Montréal, also inspired the group. Granville’s paper, “Prime Number Patterns,” was published in the April edition of American Mathematical Monthly.

Ironically, Granville just gave a lecture on “Patterns in the Primes” as part of the Distinguished Lecture Series of the Mathematical Association of America (MAA). This series, sponsored by the National Security Agency, took place at the MAA Carriage House Conference Center in Washington D.C. on Thursday, Nov. 13. This was the day before the Westfield State group’s discovery.

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The above story is based on materials provided by **Westfield State College**. *Note: Materials may be edited for content and length.*

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