"Seventeen or bust" was the rallying cry. Now it's "thirteen or bust." A rag-tag group of math aficionados have had dramatic success in the past week and are well on their way to cracking a well-known conjecture in number theory: the Sierpinski Problem.
In 1960, Polish mathematician Waclaw Sierpinski discovered, surprisingly, that the expression k * 2n + 1 was never prime when certain values of k were used, no matter what natural number n was plugged into the formula. These values of k are known as Sierpinski numbers. "There's no obvious reason why they should exist," says Chris Caldwell, a mathematician at the University of Tennessee, Martin. "For example, if you look at 3 * 2n + 1, it dumps out primes quite regularly, and the obvious feeling was they all should." But Sierpinski proved that the number 78,557--and an infinite number of others now known as Sierpinski numbers--always spit out nonprimes. Furthermore, the structure of Sierpinski's proof implied that 78,557 was the smallest such number, but nobody really knew for sure whether this was the case. This is the Sierpinski Problem: Is 78,557 truly the smallest Sierpinski number?
Mathematicians knew of 17 numbers smaller than 78,557 that might be Sierpinski numbers. To narrow down the list, a team of about 1000 volunteers recently dedicated the unused processing power of their computers. Led by computer science student Louis Helm of the University of Michigan, Ann Arbor, and programmer David Norris of the University of Illinois, Urbana-Champaign, the "Seventeen or Bust" collaboration began systematically testing each of the candidates. Between 27 November and 10 December, the team had proved that four of the candidates generated prime numbers--proving that they are not Sierpinski numbers. "It was a very exciting week," says Helm. Getting four primes in rapid succession was "basically luck," says Norris, who expects that knocking down the 13 remaining Sierpinski candidates will take the better part of a decade.
Although the Seventeen or Bust group is enthusiastic about its early success, mathematicians won't be terribly moved by the ultimate triumph or failure of the team. "The world will not be different when it's proved," says Caldwell. Nevertheless, volunteers keep streaming in, recently raising the membership to more than 1600, something that surprised even Helm. "I guess people want to be part of something big and something successful."
Seventeen or Bust