ikeaSeventeen or Bust is a distributed computing project with the goal of proving that 78,557 is the lowest Sierpinski Number. The project was started in March of 2002 as a collaboration between Louis Helm at the University of Michigan and David Norris at the University of Illinois.
A Sierpinski Number is an odd natural number k such that k2n + 1 produces non-primes for all natural numbers n (1, 2, ... ∞).
John Selfridge proved that 78557(2n) + 1, always produces composites (i.e. non-primes) and is thus a Sierpinski number; he guessed that it is probably the smallest (i.e. first) such number.
There are two ways to decide this:
1.) find a proof for a smaller k. To show that a number is a Sierpinski number you have to prove that it always produces composites when you plug it into the formula (this cannot be done by experimentation because the series is infinite),
2.) find at least one prime for k2n + 1 for all positive odd integers k < 78,557. (it can be shown that a number is not a Sierpinski number if even one result is a prime number).
The problem with option 1 is that no such proof has been found. The problem with option 2 is that you could go on forever looking for a prime number in the series if the k in question is in fact a Sierpinski number.
When the Seventeen or Bust project started, there were 17 such ks left where some n had been found where k2n + 1 produced a prime number for all but 17 numbers less than 78557. However, if a prime is found for the remaining ks then you've done it—hence the name Seventeen or Bust; you can't ever be sure of completing the search—if you can't find primes for all 17, you will not prove that 78557 is the smallest Sierpinski number, since you have no proof that the remaining k is/are in fact Sierpinski number(s); not finding a prime doesn't tell you whether or not one exists.
Seventeen or Bust is looking for an n where k2n + 1 is prime for each of the remaining 17 ks. That means trying out a lot of ns for each of the ks left to try to find a prime number.
To prove that p is a prime number, every possible prime divisor that's less than the square root of p must be checked; this means a lot of calculations for big ns (although this is probably not strictly accurate, as there are some clever mathematical tricks that refine the search). In addition, the n's are big—some of the primes found by Seventeen or Bust have half a million digits in decimal notation.
Volunteers participate in Seventeen or Bust by donating some of their CPU time to do these calculations. The client application gets work instructions from the central server, processes the work, sends back the result, and gets the next bit of work.
So far 11 of these 17 candidates have been eliminated by finding some value n where a prime number was generated by the formula.
These numbers were: 4847*, 5359*, 10223, 19249*, 21181, 22699, 24737, 27653*, 28433*, 33661*, 44131*, 46157*, 54767*, 55459, 65567*, 67607, 69109*
What is the next known Sierpinski number?