Seventeen or Bust

Short description: Volunteer computing project to find the smallest number k for which k2^n+1 is always composite

Seventeen or Bust is a volunteer computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem. The project solved eleven cases before a server loss in April 2016 forced it to cease operations. Work on the Sierpinski problem moved to PrimeGrid, which solved a twelfth case in October 2016. Five cases remain unsolved (As of January 2024).

Goals

Seventeen or Bust old client

The goal of the project is to prove that 78557 is the smallest Sierpinski number, that is, the least odd k such that k·2ⁿ+1 is composite (i.e. not prime) for all n > 0. When the project began, there were only seventeen values of k < 78557 for which the corresponding sequence was not known to contain a prime.

For each of those seventeen values of k, the project searches for a prime number in the sequence

k·2¹+1, k·2²+1, …, k·2ⁿ+1, …

testing candidate values n using Proth's theorem. If one is found, it proves that k is not a Sierpinski number. If this is done for all seventeen values, the conjectured answer 78557 to the Sierpinski problem will be proven true.

There is also the possibility that some of the sequences contain no prime numbers. In that case, the search will continue forever, searching for prime numbers where none can be found. However, there is empirical evidence suggesting the conjecture is true.

Every known Sierpinski number k has a small covering set, a finite set of primes with at least one dividing k·2ⁿ+1 for each n>0 (or else k has algebraic factorizations for some n values and a finite prime set that works only for the remaining n). For example, for the smallest known Sierpinski number, 78557, the covering set is {3,5,7,13,19,37,73}. By considering the possible values of n modulo 36, it can be shown that one of these primes is always a factor. For another known Sierpinski number, 271129, the covering set is {3,5,7,13,17,241}. Each of the remaining sequences has been tested and none has a small covering set, so it is suspected that each of them contains primes.

The second generation of the client was based on Prime95, which is used in the Great Internet Mersenne Prime Search. In January 2010, the Seventeen or Bust project started collaboration with PrimeGrid which uses the software LLR for its tests related to the Sierpinski problem.

The Seventeen or Bust server went down during April 2016, when the server and backups were lost for reasons that were not revealed to the public. The independent project is no longer active, but work on the problem continues at PrimeGrid under the same name.

Progress of the search

Twelve prime numbers have been found to date, eleven by the original Seventeen or Bust, and a twelfth by PrimeGrid's SoB project:

k	n	Digits of k·2n+1	Date of discovery	Found by
46,157	698,207	210,186	26 Nov 2002	Stephen Gibson
65,567	1,013,803	305,190	03 Dec 2002	James Burt
44,131	995,972	299,823	06 Dec 2002	deviced (nickname)
69,109	1,157,446	348,431	07 Dec 2002	Sean DiMichele
54,767	1,337,287	402,569	22 Dec 2002	Peter Coels
5,359	5,054,502	1,521,561	06 Dec 2003	Randy Sundquist
28,433	7,830,457	2,357,207	30 Dec 2004	Anonymous
27,653	9,167,433	2,759,677	08 Jun 2005	Derek Gordon
4,847	3,321,063	999,744	15 Oct 2005	Richard Hassler
19,249	13,018,586	3,918,990	26 Mar 2007	Konstantin Agafonov
33,661	7,031,232	2,116,617	13 Oct 2007	Sturle Sunde
10,223	31,172,165	9,383,761	31 Oct 2016	Péter Szabolcs
21,181	≳ 40,000,000	≳ 12,043,495	(Search in progress)
22,699	≳ 40,800,000	≳ 12,296,072	(Search in progress)
24,737	≳ 40,900,000	≳ 12,325,559	(Search in progress)
55,459	≳ 39,800,000	≳ 11,993,490	(Search in progress)
67,607	≳ 40,200,000	≳ 12,111,889	(Search in progress)

The largest of these primes, 10223·2^31172165+1, held the record as the largest known prime number that is not a Mersenne prime from October 2016 until May 2023. The primes on this list over one million digits in length are the six known "Colbert numbers" whimsically named after Stephen Colbert. These are defined as primes which eliminate a remaining Sierpinski number candidate.

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

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

In March 2017, n had exceeded 31,000,000 for the last five k values. At that time, PrimeGrid decided to suspend testing to do a double check of all those smaller n values for which the Proth test residue had been lost, or for which the result had not been successfully verified by two independent computations on different computers. Testing resumed after the double check was finally completed on October 10, 2019, taking about two and a half years.

The current status for the remaining multipliers can be seen at PrimeGrid's website.

Modular restrictions

Every multiplier has modular restrictions for the exponent n, assuming the latter exists. For example, for k = 21,181, it is sufficient to check only values of n congruent to 20 (mod 24); the covering set for all other terms is {3, 5, 7, 13, 17}. Similarly, for k = 22,699, only terms with n congruent to 46 (mod 72) are candidates, as the set of all other terms have covering set {3, 5, 7, 13, 17, 19, 73}.

See also

• Riesel Sieve, a related volunteer computing project for numbers of the form k·2ⁿ−1
• List of volunteer computing projects
• PrimeGrid, biggest search for primes.
• Computer-assisted proof

External links

• Seventeen or Bust homepage (historical) from Internet Archive
• PrimeGrid
• A258074 Table read by rows: each row represents the constant and exponent of a Colbert number.