Sierpiński conjecture

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

Theorem: There exist infinitely many odd integers $k$ such that $k{2}^n+1$ is composite for every integer $n\ge 1$.

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 $k{2}^n+1$ for each .

Title	Sierpiński conjecture
Canonical name	SierpinskiConjecture
Date of creation	2013-03-22 13:34:16
Last modified on	2013-03-22 13:34:16
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	12
Author	yark (2760)
Entry type	Conjecture
Classification	msc 11B83
Synonym	Sierpinski conjecture