Existence of at least one prime for all sequences in the family of sequences

Question

Prove or disprove that for a fixed $n \in N$, there exists at least one prime among the integers of the form $2^{k}n+2^k-1$ for an arbitrary $k \in N$.

Answer 1

There exist infinitely many values of $n$ for which $2^k n+2^k-1 = 2^k(n-1)-1$ is composite for every positive integer $k$. The integers $n-1$ with this property are called Riesel numbers (related to Sierpinski numbers). The smallest known Riesel number is $n-1= 509$,$203$. Proving that this is a Riesel number is related to covering systems of congruences.