## Sierpinski numberA positive, odd integer k such that k times 2^{n}
+ 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 2^{n}
+ 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(2^{n} + 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 | |||||

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