Definition:Sierpiński Number of the Second Kind

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 members of the set:
 * $\left\{{k 2^n + 1}\right\}$

are composite.

The list 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.

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

It has been proven that there exist infinitely many numbers which are simultaneously Sierpiński, Riesel, and Carmichael.

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 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.

Also see

 * Definition:Riesel Number
 * Sierpiński Problem

Historical Note
He proved in 1960 that there is an infinite number of Sierpiński numbers.