A positive number k is a Sierpinski number if there
are no primes of the form
k.2n+1 for any positive
number n (for k < 2n).
Primes of this form are called Proth primes, they are relatively
easy to prove prime because of Proth's Theorem. It is much harder to
prove that a number is a Sierpinski number.
The lowest known Sierpinski number is 78557. The Sierpinski
conjecture states that this is the lowest Sierpinski number, but this
remains unproven.