Seventeen or Bust
Encyclopedia
Seventeen or Bust is a distributed computing
project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.
(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
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 conjecture
d 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.
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
.
the largest of these primes, 19249·213018586+1, is the largest known prime that is not a Mersenne prime
.
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:
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...
project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.
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 compositeComposite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
(i.e. not prime
Prime
A prime is a natural number that has exactly two distinct natural number divisors: 1 and itself.Prime or PRIME may also refer to:In mathematics:*Prime , the ′ mark, typically used as a suffix...
) 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
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
- k·21+1, k·22+1, …, k·2n+1, …
using Proth's theorem
Proth's theorem
In number theory, Proth's theorem is a primality test for Proth numbers.It states that if p is a Proth number, of the form k2n + 1 with k odd and k In number theory, Proth's theorem is a primality test for Proth numbers....
, 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 conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
d 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.
Every known Sierpinski number k has a small covering set
Covering set
In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the 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
The second generation of the client is based on Prime95
Prime95
Prime95 is the name of the Microsoft Windows-based software application written by George Woltman that is used by GIMPS, a distributed computing project dedicated to finding new Mersenne prime numbers....
, which is used in the Great Internet Mersenne Prime Search
Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search is a collaborative project of volunteers who use freely available computer software to search for Mersenne prime numbers. The project was founded by George Woltman, who also wrote the software Prime95 and MPrime for the project...
.
Prime number discoveries
The Seventeen or Bust set, with data for the eleven prime numbers eliminated to date:| # | 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 | > 15,900,000 | (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 |
the largest of these primes, 19249·213018586+1, is the largest known prime that is not a Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
.
Note that each of these numbers has enough digits to fill up a medium-sized novel
Novel
A novel is a book of long narrative in literary prose. The genre has historical roots both in the fields of the medieval and early modern romance and in the tradition of the novella. The latter supplied the present generic term in the late 18th century....
, 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
- Riesel SieveRiesel SieveRiesel Sieve is a distributed computing project, running in part on the BOINC platform. Its aim is to prove that is the smallest Riesel number, by finding a prime of the form for all odd smaller than .-Progress of the project:...
, a related distributed computing project for numbers of the form k·2n−1 - List of distributed computing projects
- PrimeGridPrimeGridPrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...
- biggest search for primes. - Computer-assisted proofComputer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer.Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and...