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Sierpinski number





A positive, odd integer k such that k times 2n + 1 is never a prime number for any value of n. In 1960 Waclaw Sierpinski showed that there were infinitely many such numbers (though he didn't give a specific example.) This a strange result. Why should it be that while the vast majority of expressions of the form m times 2n + 1 eventually produce a prime, some don't? For now, mathematicians are focused on a more manageable problem posed by Sierpinski: What is the smallest Sierpinski number? In 1962, John Selfridge discovered what is still the smallest known Sierpinski number, k = 78557. The next largest is 271129. Is there a smaller Sierpinski number? No one yet knows. However, to establish that 78557 is really the smallest, it would be sufficient to find a prime of the form k(2n + 1) for every value of k less than 78557. In early 2001, there were only 17 candidate values of k left to check: 4847, 5359, 10223, 19249, 21181, 22699, 24737, 27653, 28433, 33661, 44131, 46157, 54767, 55459, 65567, 67607, and 69109. In March 2002, Louis Helm of the University of Michigan and David Norris of the University of Illinois started a project called "Seventeen or Bust," the goal of which is to harness the computing power of a worldwide network of hundreds of personal computers to check for primes among the remaining candidates. The team's effort have so far eliminated five candidates – 46157, 65567, 44131, 69109, and 54767. Despite this encouraging start, it may take as long as a decade, with many additional participants, to check the dozen remaining candidates.


Related category

   • TYPES OF NUMBERS