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# Sierpiński conjecture

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

Theorem: 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. 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 $k<78557$.

Synonym:

Sierpinski conjecture

Type of Math Object:

Conjecture

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

11B83*no label found*

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## Comments

## Other "Sierpinski conjectures"

There is a variant of the ErdÃ¶s-Straus conjecture also referred to as "Sierpinski conjecture", namely: For all n>1, 5/n can be written as sum of 3 unit fractions ; published by W.S. in 1956 (cf. Wikipedia and http://mathworld.wolfram.com/EgyptianFraction.html).

See also http://mathworld.wolfram.com/SierpinskisConjecture.html for yet another "Sierpinski's conjecture".