Selfridge's conjecture

In number theory, a Sierpiński number is an odd natural number k such that $k\times 2^{n}+1$ is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all members of the following set are composite: $\left\{\,k\cdot 2^{n}+1:n\in \mathbb {N} \,\right\}.$ If the form is instead $k\times 2^{n}-1$, then k is a Riesel number.