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Westfield State College closes a 'GAP' in understanding primes |
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By Katelyn Gendron
Reminder Assistant Editor
WESTFIELD On Nov. 14, a four-man team of Westfield State College (WSC) mathematics professors and undergraduate students made a new discovery in their field.
For several weeks, Professor Julian Fleron, PhD., Professor Marcus Jaiclin, PhD., and senior mathematic majors Michael Guenette and Jeffrey Vanasse worked to discover the first known example of a 3 by 3 by 3 generalized arithmetic progression (GAP).
In an interview with Reminder Publications, Fleron explained that the task was evident to the group after reading "Prime Number Patterns," a paper by mathematician Andrew Granville, in which he states that he was never able to find a 3 by 3 by 3 GAP.
"We wanted to see how hard it would be to find one," Guenette said. "There are existance proofs but they don't give specific examples [of 3 by 3 by 3 GAPs]."
Fleron explained that this concept is "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 [and] ends with 27,917 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."
Jaiclin explained that in order to discover the GAP he had to create an algorithm within the Linux computer operating system in C++ language. The program allowed the team to generate millions of possible 3 by 3 by 3 GAPs.
"We were always optimistic, but the first tests got us really excited that our method would be successful," Vanasse said.
Fleron noted that since the initial discovery, the team has found 19 additional 3 by 3 by 3 GAPs.
Guenette said the team is committed to finding as many examples of 3 by 3 by 3 GAPs as they can, which he hopes will also provide them with more information about primes as well.
"[This discovery] is important to Westfield State because we want our undergraduates to experience what it's like to do original mathematics," Jaiclin said. "Most undergraduates don't get that experience."
Fleron explained that his students were also exposed to the work of 33-year-old mathematician and number theorist Terence Chi-Shen Tao.
"Many prominent number theorists are working simply to understand the implications of these discoveries [in number theory]," Fleron said.
He explained that number theory and the "effort to understand prime numbers is one of the greatest quests in all of mathematics."
"The prime numbers are the building blocks for everything we know about whole numbers, like the periodic table is for chemistry," Fleron said. "The effort to understand prime numbers, if it's ever won by humanity, will be won by foot soldiers like us and one general like Tao."
Fleron said he believes their discovery in finding the highest dimensional GAP will remain uncontested for many years as the largest prime in a 4 by 4 by 4 or a 3 by 3 by 3 GAP is about 5,000,000,000,000,000 [five quadrillion].
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