# Sierpiński conjecture

In 1960 Wacław Sierpiński (1882-1969) proved the following interesting result:

There exist infinitely many odd integers $k$ such that $k2^{n}+1$ is composite for every integer $n\geq 1$.

A multiplier $k$ with this property is called a Sierpiński number (http://planetmath.org/SierpinskiNumbers). The Sierpiński problem consists in determining the smallest Sierpiński number. In 1962, John Selfridge discovered the Sierpiński number $k=78557$, which is now believed to be in fact the smallest such number.

Conjecture: The integer $k=78557$ is the smallest Sierpiński number.

To prove the conjecture, it would be sufficient to exhibit a prime $k2^{n}+1$ for each $k<78557$.

Title Sierpiński conjecture SierpinskiConjecture 2013-03-22 13:34:16 2013-03-22 13:34:16 yark (2760) yark (2760) 12 yark (2760) Conjecture msc 11B83 Sierpinski conjecture