On generalized Mersenne prime

Abstract

The prime numbers of the form \(M_{p,q} =p^q-p+1,\) where \(p\) and \(q\) are positive integers, we call generalized Mersenne primes (GMPs). In this paper we study GMPs and establish some important results based on these primes in connection with some arithmetic functions. We also discuss solvability of quadratic congruences modulo \(2q+1\) under certain condition on \(M_{p,q}\). We also show that a certain type of these primes generate a family of imaginary quadratic fields whose class numbers are multiple of 3. We also derive an irreducibility criterion of an algebraic integer in the real quadratic field \({\mathbb {Q}}\left( {\sqrt{9({M_{p,q}+2})^2+12}}\right) \).