On Prime Primitive Roots of $2^{k}p+1$

Abstract

A prime $p$ is a Sophie Germain prime if $2p+1$ is prime as well. An integer $a$ that is coprime to a positive integer $n>1$ is a primitive root of $n$ if the order of $a$ modulo $n$ is $\phi(n).$ Ramesh and Makeshwari proved that, if $p$ is a prime primitive root of $2p+1$, then $p$ is a Sophie Germain prime. Since there exist primes $p$ that are primitive roots of $2p+1$, in this note we consider the following general problem: For what primes $p$ and positive integers $k>1$, is $p$ a primitive root of $2^{k}p+1$? We prove that it is possible only if $(p,k)\in \{(2,2), (3,3), (3,4), (5,4)\}.$