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.