# Seventeen or Bust

Seventeen or Bust was a distributed 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.[1] Five cases remain unsolved as of April 2020.[2]

## Goals

Seventeen or Bust old client

The goal of the project was 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 searched 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 was found, it proved that k was not a Sierpinski number. If the goal had been reached, the conjectured answer 78557 to the Sierpinski problem would be proven true.

There is also the possibility that some of the sequences contain no prime numbers. In that case, the search would continue forever, searching for prime numbers where none can be found. However, there is some empirical evidence suggesting the conjecture is true.[3]

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[4]). For example, for the smallest known Sierpinski number, 78557, the covering set is {3,5,7,13,19,37,73}. 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 Sierpiński problem.[2]

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 The project is no longer active. Work on the Sierpinski problem continues at PrimeGrid.[5][6]

## 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:[2]

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[7][1] Péter Szabolcs
21,181 ≳ 32,000,000 ≳ 9,632,964 (Search in progress)
22,699 ≳ 32,000,000 ≳ 9,632,964 (Search in progress)
24,737 ≳ 32,000,000 ≳ 9,632,964 (Search in progress)
55,459 ≳ 32,000,000 ≳ 9,632,964 (Search in progress)
67,607 ≳ 32,000,000 ≳ 9,632,964 (Search in progress)

As of April 2020 the largest of these primes, 10223·231172165+1, is the largest known prime number that is not a Mersenne prime.[8] 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.[9][10]

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.[11] Testing resumed after the double check was completed on October 10, 2019, taking about two and a half years.[12]

The current status for the remaining multipliers can be seen at PrimeGrid's website.[13]

## 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 22 (mod 24) are candidates, as the set of all other terms have covering set {3, 5, 7, 13, 17}.

## References

1. ^ a b "PrimeGrid's Seventeen or Bust Subproject, Official Announcement" (PDF). 2016.
2. ^ a b c Michael Goetz. "Seventeen or Bust and the Sierpinski Problem (PrimeGrid Forum)".
3. ^ Chris Caldwell. "Sierpinski number".
4. ^
5. ^ Michael Goetz. "Re: Server down?". Archived from the original on 28 June 2016.
6. ^ Michael Goetz. "Re: Update on seventeenorbust.com".