If $p$ is prime, does there exist a positive integer $k$ such that $2^k\,p+1$ is also prime?
Actually $\lambda$ might not be finite, for example $271129$ is prime and $271129 \cdot 2^k+1$ is never prime. This is a special case of a Sierpinski number. Every number in the set $\{271129 \cdot 2^k+1\}$ is divisible by a number in the set $\{3, 5, 7, 13, 17, 241\}$.