Sierpinski problem

In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that ${\displaystyle k\times 2^{n}+1}$ is composite, for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.

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

Numbers in such a set with odd k and k < 2ⁿ are Proth numbers.

The sequence of currently known Sierpiński numbers begins with:

The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form 78557⋅2ⁿ + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. All currently known Sierpiński numbers possess similar covering sets.

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

In 1967, Sierpiński and Selfridge conjectured that 78,557 is the smallest Sierpiński number, and thus the answer to the Sierpiński problem.

To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are not Sierpiński numbers. That is, for every odd k below 78,557 there exists a positive integer n such that k2ⁿ+1 is prime. As of November 2016^[update], there are only five candidates:

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