On the smallest $k$ such that all $k\cdot 2^{n}+1$ are composite

Abstract:

In this note we present some computational results which restrict the least odd value of k such that $k \cdot {2^n} + 1$ is composite for all $n \geqslant 1$ to one of 91 numbers between 3061 and 78557,inclusive. Further, we give the computational results of a relaxed problem and prove for any positive integer r the existence of infinitely many odd integers k such that $k\cdot {2^r} + 1$ is prime but $k\cdot {2^v} + 1$ is not prime for $v < r$.