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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. Five cases remain unsolved as of March 2017. Work on the Sierpinski problem is now being done at PrimeGrid.
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Goals
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.
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 was based on Prime95, which is used in the Great Internet Mersenne Prime Search.
The Seventeen or Bust server went down on April, 2016, the server and backups were lost. The project is no longer active. Work on the Sierpinski problem continues at PrimeGrid.
Progress of the search
Seventeen or Bust, including PrimeGrid's SoB project, has found twelve prime numbers to date:
As of November 2016 the largest of these primes, 10223·231172165+1, is the largest known prime number that is not a Mersenne prime. The primes on this list over one million digits in length are the six known Colbert Numbers.
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 five remaining sequences:
k·2n+1, for k = 21181, 22699, 24737, 55459, 67607.In March 2017, n had exceeded 31,000,000 for these five k values. At that time, PrimeGrid decided 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. The double check is expected to take years.