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 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.
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.
Read more about this topic: Seventeen Or Bust
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—Frances Kuehn (b. 1943)
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—Anton Pavlovich Chekhov (1860–1904)