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Sierpinski number

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In mathematics, a Sierpinski number is a positive, odd, number k such that integers of the form k2n + 1 are composite (i.e. not prime) for all natural numbers n.

In other words, when k is a Sierpinski number, all members of the following set are composite:

<math>\left\{\,k 2^n + 1 : n \in\mathbb{N}\,\right\}</math>

In 1960 Waclaw Sierpinski proved that there are an infinite number of odd integers that when used as k produce no primes.

The Sierpinski Problem is: "What is the smallest Sierpinski number?"

In 1962, John Selfridge[?] proposed that 78,557 was the answer to the Sierpinski problem. Selfridge found that when 78,557 was used as k in the equation, none of the numbers produced by the equation were prime. In other words, Selfridge demonstrated that 78,557 is a Sierpinkski number.

Seventeen or Bust, a distributed computing project, is trying all untested numbers that are less than 78,557 to see whether they are Sierpinski numbers. If the project finds that all of these numbers do generate a prime number when used as k, the project will have found a proof to Selfridge's conjecture. When the project started there were 17 numbers k for which such a prime was unknown (hence the name of the project) and in the first year of its existence the project succeeded in finding 5 more primes; hence (as of April 2003) there are 12 more ks to be tested.

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