The Sierpinski conjecture states that the lowest Sierpinski number is 78557.
It has been proven that that number is a Sierpinski number, but not
that it is the lowest. As of 4 Jan 2003, twelve lower candidates
remain: 4847, 5359, 10223, 19249, 21181, 22699, 24737, 27653, 28433,
33661, 55459, and 67607. For all other numbers below 78557, it has
been proven that they are not Sierpinski numbers. For these twelve,
there are reasons to believe they're not Sierpinski numbers (but no
proof yet, of course). These arguments are over my head.
Seventeen or Bust is a distributed computing project to prove
the conjecture. At the time it started, there were seventeen
candidates left, so it may just get there. What needs to be done is to
find a n for each of these numbers k, so that
k.2n+1 is prime.