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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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## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

Jun 6

new question: difference of a function and a finite sum by pfb

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

Jun 6

new question: difference of a function and a finite sum by pfb

## Corrections

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Alternate spelling by rm50 ✓

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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".