A Sierpiński number of the second kind is a number satisfying Sierpiński's composite number theorem, i.e., a Proth number such that is composite for every .
The smallest known example is , proved in 1962 by J. Selfridge, but the fate of a number of smaller candidates remains to be determined before this number can be established as the smallest such number. As of 1996, 35 candidates remained (Ribenboim 1996, p. 358), a number which had been reduced to 17 by the beginning of 2002 (Peterson 2003).
In March 2002, L. K. Helm and D. A. Norris began a distributed computing effort dubbed "seventeen or bust" to eliminate the remaining candidates. With the aid of collaborators across the globe, this number was reduced to 12 as of December 2003 (Peterson 2003, Helm and Norris). The following table summarizes numbers subsequently found to be prime by "seventeen or bust," leaving only five candidates remaining as of November 2016.
date | participant | number |
Dec. 6, 2003 | ||
Jun. 8, 2005 | D. Gordon | |
Oct. 15, 2005 | R. Hassler | |
May 5, 2007 | K. Agafonov | |
Oct. 30, 2007 | S. Sunde | |
Nov. 6, 2016 | P. Szabolcs |
The following table lists the known primes together with the only remaining candidates which, as Jan. 2008, are the six numbers 10223, 21181, 22699, 24737, 55459, and 67607. A list of primes found by the project is also maintained by Caldwell (http://primes.utm.edu/bios/page.php?id=429).
prime | digits | Caldwell | |
4847 | 999744 | http://primes.utm.edu/primes/page.php?id=75994 | |
5359 | 1521561 | http://primes.utm.edu/primes/page.php?id=67719 | |
10223 | 9383761 | http://primes.utm.edu/primes/page.php?id=122473 | |
19249 | 3918990 | http://primes.utm.edu/primes/page.php?id=80385 | |
21181 | |||
22699 | |||
24737 | |||
27653 | 2759677 | http://primes.utm.edu/primes/page.php?id=74836 | |
28433 | 2357207 | http://primes.utm.edu/primes/page.php?id=73145 | |
33661 | 2116617 | http://primes.utm.edu/primes/page.php?id=82804 | |
44131 | 299823 | http://primes.utm.edu/primes/page.php?id=62867 | |
46157 | 210186 | http://primes.utm.edu/primes/page.php?id=62865 | |
54767 | 402569 | http://primes.utm.edu/primes/page.php?id=62869 | |
55459 | |||
65567 | 305190 | http://primes.utm.edu/primes/page.php?id=62866 | |
67607 | |||
69109 | 348431 | http://primes.utm.edu/primes/page.php?id=62868 |
Consider now restricting Sierpiński numbers of the second kind to those with prime . The smallest proved prime Sierpiński number is 271129. A distributed computing project to find examples of that are prime with smaller than the proven lower limit is currently underway (Caldwell). Note that the smallest candidates include three prime candidates from the "seventeen or bust" list: 10223, 22699, 67607. A list of primes found by the project is maintained by Caldwell (http://primes.utm.edu/bios/page.php?id=564).
Let be smallest for which is prime, then the first few values are 0, 1, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, ... (OEIS A046067). The second smallest are given by 1, 2, 3, 4, 2, 3, 8, 2, 15, 10, 4, 9, 4, 4, 3, 60, 6, 3, 4, 2, 11, 6, 9, 1483, ... (OEIS A046068). Quite large can be required to obtain the first prime even for small . For example, the smallest prime of the form is .
There are an infinite number of Sierpiński numbers which are prime.
The smallest odd such that is composite for all are 773, 2131, 2491, 4471, 5101, ... (OEIS A033919).
is always composite for and Gaussian integers , , and . (E. Pegg Jr., pers. comm., Feb. 6, 2003; Broadhurst 2005).