Seventeen or Bust
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This article relies on references to primary sources. (February 2008) |
Seventeen or Bust is a distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.
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Goals [edit]
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 is 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, …
using Proth's theorem, 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 in the remaining six. 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 remaining 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.[1]
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}. None of the remaining sequences has a small covering set (that can be easily tested) 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.
Prime number discoveries [edit]
The Seventeen or Bust set, with data for the eleven prime numbers eliminated to date:[2]
| # | k | n | Digits of k·2n+1 | Date of discovery | Found by |
|---|---|---|---|---|---|
| 1 | 4,847 | 3,321,063 | 999,744 | 15 Oct 2005 | Richard Hassler |
| 2 | 5,359 | 5,054,502 | 1,521,561 | 06 Dec 2003 | Randy Sundquist |
| 3 | 10,223 | > 17,000,000 | (Search in progress) | ||
| 4 | 19,249 | 13,018,586 | 3,918,990 | 26 Mar 2007 | Konstantin Agafonov |
| 5 | 21,181 | > 17,000,000 | (Search in progress) | ||
| 6 | 22,699 | > 17,000,000 | (Search in progress) | ||
| 7 | 24,737 | > 17,000,000 | (Search in progress) | ||
| 8 | 27,653 | 9,167,433 | 2,759,677 | 08 Jun 2005 | Derek Gordon |
| 9 | 28,433 | 7,830,457 | 2,357,207 | 30 Dec 2004 | Anonymous |
| 10 | 33,661 | 7,031,232 | 2,116,617 | 13 Oct 2007 | Sturle Sunde |
| 11 | 44,131 | 995,972 | 299,823 | 06 Dec 2002 | deviced (nickname) |
| 12 | 46,157 | 698,207 | 210,186 | 26 Nov 2002 | Stephen Gibson |
| 13 | 54,767 | 1,337,287 | 402,569 | 22 Dec 2002 | Peter Coels |
| 14 | 55,459 | > 17,000,000 | (Search in progress) | ||
| 15 | 65,567 | 1,013,803 | 305,190 | 03 Dec 2002 | James Burt |
| 16 | 67,607 | > 17,000,000 | (Search in progress) | ||
| 17 | 69,109 | 1,157,446 | 348,431 | 07 Dec 2002 | Sean DiMichele |
As of December 2011[update] the largest of these primes, 19249·213018586+1, is the largest known prime number that is not a Mersenne prime.[3]
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 each of the six remaining sequences:
- k·2n+1, for k = 10223, 21181, 22699, 24737, 55459, 67607.
See also [edit]
- 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 [edit]
- ^ Chris Caldwell. "Sierpinski number".
- ^ Seventeen or Bust: Project Stats
- ^ "The Largest Known Primes--A Summary".
