Definition:Sierpiński Number of the Second Kind

Definition
A Sierpiński number 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, all members of the set:
 * $\left\{{k 2^n + 1}\right\}$

are composite.

The list of known Sierpiński numbers 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.

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 see

 * Sierpiński Problem

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