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 \((p,k)\in \{(2,2), (3,3), (3,4), (5,4)\}.\)