Definition:Sierpiński Number
From ProofWiki
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:
- $78\ 557, \ 271\ 129, \ 271\ 577, \ 322\ 523, \ 327\ 739, \ 482\ 719, \ 575\ 041, \ 603\ 713, \ 903\ 983, \ 934\ 909, \ 965\ 431, \ \ldots$
This sequence is A076336 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
It has been conjectured that $78\ 557$ is the smallest Sierpiński number.
It was proved by John Selfridge in 1962 that $78\ 557$ is Sierpiński, but there are still some numbers smaller than that whose status is uncertain.
Also see
Source of Name
This entry was named for Wacław Sierpiński.
He proved in 1960 that there is an infinite number of Sierpiński numbers.
