Seventeen or Bust

From Wikipedia, the free encyclopedia

This article or section needs sources or references that appear in reliable, third-party publications.
Primary sources and sources affiliated with the subject of the article are generally not sufficient for a Wikipedia article. Please include more appropriate citations from reliable sources, or discuss the issue on the talk page.
This article has been tagged since February 2008.

Seventeen or Bust logo

Seventeen or Bust is a distributed computing project to solve the last seventeen cases in the Sierpinski problem.

The goal of the project is to prove that 78,557 is the smallest Sierpinski number, that is, the least odd k such that k·2ⁿ+1 is composite for all n > 0 (i.e. not prime for any n). When the project began, there were only seventeen values of k < 78,557 that were still in question.

For each of those seventeen values of k, the project is searching for a value of n for which k·2ⁿ+1 is prime, thereby proving that k is not a Sierpinski number. So far, the project has found prime numbers in eleven of the sequences, and is continuing to search the remaining six. If the goal is reached, the conjecture of the Sierpinski problem will be proven true.

There is also the possibility that some of the remaining sequences contain no prime numbers; if that possibility weren't present, the problem would not be interesting. In that case, the search would continue forever, searching for prime numbers where none can be found. However, there is some empirical evidence that the conjecture is true. ^[1] Each of the known Sierpinski numbers has a small covering set, a set of numbers where at least one of them divides each k·2ⁿ+1. For example, for the smallest known Sierpinski number (78,557) the covering set is {3,5,7,13,19,37,73}. For the next number, 271129, the covering set is {3,5,7,13,17,241}. None of the remaining sequences have a small covering set (a much easier test) so it is suspected that they all have counter-examples, too.

The eleven prime numbers found so far by the project are:

#	k	n	Digits of k·2ⁿ+1	Date of discovery	Who found
1°	4,847	3,321,063	999,744	October 15, 2005	Richard Hassler
2°	5,359	5,054,502	1,521,561	December 6, 2003	Randy Sundquist
3°	10,223
4°	19,249	13,018,586	3,918,990	March 26, 2007	Konstantin Agafonov
5°	21,181
6°	22,699
7°	24,737
8°	27,653	9,167,433	2,759,677	June 8, 2005	Derek Gordon
9°	28,433	7,830,457	2,357,207	December 30, 2004	Anonymous
10°	33,661	7,031,232	2,116,617	October 13, 2007	Sturle Sunde
11°	44,131	995,972	299,823	December 6, 2002	deviced (nickname)
12°	46,157	698,207	210,186	November 26, 2002	Stephen Gibson
13°	54,767	1,337,287	402,569	December 22, 2002	Peter Coels
14°	55,459
15°	65,567	1,013,803	305,190	December 3, 2002	James Burt
16°	67,607
17°	69,109	1,157,446	348,431	December 7, 2002	Sean DiMichele

As of March 2008 the largest of these primes, 19249·2^13018586+1, is the largest known prime that is not a Mersenne prime.^[2]

Note that each of these numbers has enough digits to fill up a medium-sized novel, at least. The project is presently dividing numbers among its active users, in hope of finding a prime number in the six remaining sequences:

k·2ⁿ+1, for k = 10223, 21181, 22699, 24737, 55459, 67607.