November 25, 2008
A team of students has identified the first known example of a particular three-dimensional array of numbers that consists of 27 primes. Mathematics majors Jeffrey P. Vanasse and Michael E. Guenette, working under the direction of Marcus Jaiclin and Julian F. Fleron of Westfield State College in Massachusetts, made the discovery.
The discovery is a 3-by-3-by-3 generalized arithmetic progression consisting of only prime numbers. An arithmetic progression is a sequence of numbers such that the difference between successive terms is a constant. A generalized (or multidimensional) arithmetic progression (GAP) allows for several possible differences.
The newly discovered array consists of 27 primes, with 929 as its smallest prime and 27917 as its largest. The 25 intervening primes are constructed by adding combinations of the numbers 2904, 3150, and 7440 in an appropriately structured way.
"Such an object was known to exist and its approximate size had been loosely estimated," Fleron said. However, a blind search would have necessitated "checking more cases than can be feasibly checked by all existing modern computers, each running for the next million years," he observed.
Instead, Fleron noted, the team's understanding of the structural relationships between the potential prime candidates allowed them to winnow down the possibilities. The students then developed a computer algorithm to search for the required pattern.
Still, Guenette said, "we were worried that it might take months to run based on our estimates."
"We were always optimistic," fellow student Vanasse indicated, "but the first tests got us really excited that our method would be successful." Their computer algorithm came up with the answer on Nov. 14.
The team was inspired by the work of Terence Tao of the University of California, Los Angeles and Andrew Granville of the Université de Montréal. Granville had written about generalized arithmetic progressions of primes in his article "Prime Number Patterns," which was published in the April issue of The American Mathematical Monthly. "We have been unable to find a 3-by-3-by-3 GAP of distinct primes," Granville had noted in his article.
"Many prominent number theorists are working simply to understand the implications of these discoveries," Fleron stated. "Now Westfield State College students are playing a role, as well."
Source: Westfield State College, Nov. 17, 2008.