Seventeen or Bust
Seventeen or Bust is a distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem. The project has solved eleven cases, and continues to search for solutions to the remaining six.[1]
Goals
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 is searching 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, that proves k is not a Sierpinski number. If the goal is reached, 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 would continue forever, searching for prime numbers where none can be found. However, there is some empirical evidence suggesting the conjecture is true.[2]
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. 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 is based on Prime95, which is used in the Great Internet Mersenne Prime Search.
Progress of the search
Seventeen or Bust has found eleven prime numbers to date:[1]
| 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 | > 29,514,665 | > 8,884,803 | (Search in progress) | |
| 21,181 | > 27,700,000 | > 8,338,535 | (Search in progress) | |
| 22,699 | > 27,700,000 | > 8,338,535 | (Search in progress) | |
| 24,737 | > 27,700,000 | > 8,338,535 | (Search in progress) | |
| 55,459 | > 27,700,000 | > 8,338,535 | (Search in progress) | |
| 67,607 | > 29,440,211 | > 8,862,391 | (Search in progress) | |
As of February 2016 the largest of these primes, 19249·213018586+1, is the largest known prime number that is not a Mersenne prime.[3] The primes on this list over one million digits in length are the five known Colbert Numbers.[4][5]
Each of these numbers has enough digits to fill up a medium-sized novel, at least. The project is dividing numbers among its active users, in hope of finding a prime number in each of the six remaining sequences:
- k·2n+1, for k = 10223, 21181, 22699, 24737, 55459, 67607.
See also
- Riesel Sieve, a related distributed computing project for numbers of the form k·2n−1
- List of distributed computing projects
- PrimeGrid - biggest search for primes.
- Computer-assisted proof
References
- 1 2 Seventeen or Bust: Project Stats
- ↑ Chris Caldwell. "Sierpinski number".
- ↑ "The Top Twenty Largest Known Primes". The Prime Pages. Retrieved 11 August 2013.
- ↑ Colbert Number - from Wolfram MathWorld. Mathworld.wolfram.com (2009-04-05). Retrieved on 2014-05-11.
- ↑ The Prime Glossary: Colbert number. Primes.utm.edu. Retrieved on 2014-05-11.