Definition:Sierpiński Number of the Second Kind

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Definition

A Sierpiński number of the second kind is an odd positive integer $k$ such that integers of the form $k2^n + 1$ are composite for all positive integers $n$.

That is, when $k$ is a Sierpiński number of the second kind, all elements of the set:

$\set {k 2^n + 1}$

are composite.


Sequence

The sequence of known Sierpiński numbers of the second kind starts:

$78\ 557, \ 271\ 129, \ 271\ 577, \ 322\ 523, \ 327\ 739, \ 482\ 719, \ 575\ 041, \ 603\ 713, \ 903\ 983, \ 934\ 909, \ 965\ 431, \ \ldots$


It has been conjectured that $78 \, 557$ is the smallest Sierpiński number of the second kind.

Although it was proved by John Selfridge in $1962$ that $78 \, 557$ is Sierpiński, there are still some numbers smaller than that whose status is uncertain.


Also known as

A Sierpiński number of the second kind is also often generally known as a Sierpiński number, as the Sierpiński numbers of the first kind have not received the same amount of attention.

However, since the philosophy of $\mathsf{Pr} \infty \mathsf{fWiki}$ is to include all and everything, it is necessary to ensure full distinction is made between the two.

Hence, whenever used, the full title will be used for this entity throughout.

However, when discussing the nature of whether a given integer $n$ is a Sierpiński number of the second kind or not, it is commonplace, even on $\mathsf{Pr} \infty \mathsf{fWiki}$, to state: $n$ is / is not Sierpiński.


Also see

  • Results about Sierpiński Numbers of the Second Kind can be found here.


Source of Name

This entry was named for Wacław Franciszek Sierpiński.


Historical Note

Wacław Franciszek Sierpiński proved in $1960$ that there is an infinite number of Sierpiński numbers of the second kind.


Sources

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