Sierpinski number

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In number theory, a Sierpinski number is an odd natural 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:

$\left\{\,k 2^n + 1 : n \in\mathbb{N}\,\right\}$

Numbers in this set with odd k and k < 2ⁿ are called Proth numbers.

In 1960 Wacław Sierpiński proved that there are infinitely many odd integers that when used as k produce no primes.

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

In 1962, John Selfridge proved that 78,557 is a Sierpinski number; he showed that, when k=78,557, all numbers of the form k2ⁿ+1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}.

In addition, he proposed (but could not prove) Selfridge's conjecture: that 78,557 is the smallest Sierpinski number, and thus the answer to the Sierpinski problem.

To show that 78,557 really is the smallest Sierpinski number, one must show that all the odd numbers smaller than 78,557 are not Sierpinski numbers. That is, there exists an n such that k2ⁿ+1 is prime. As of 2007, there are only eight candidates which have not been eliminated as possible Sierpinski numbers. Seventeen or Bust, a distributed computing project, is testing these remaining numbers. If the project finds a prime of the right form for all the remaining k, the project will have completed the proof of Selfridge's conjecture.