Sierpinski number

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In mathematics, 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 proposed what is known as Selfridge's conjecture: that 78,557 was the answer to the Sierpinski problem. Selfridge found that when 78,557 was used as k, all of the resulting set could be factored by members of the covering set {3, 5, 7, 13, 19, 37, 73}. In other words, Selfridge proved that 78,557 is a Sierpinski number.

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. As of 2006, all but eight of these numbers had been shown to produce primes, and were thus eliminated as possible Sierpinski numbers. Seventeen or Bust, a distributed computing project, is testing these remaining 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.