Mersenne Primes in Real Quadratic Fields

Abstract

The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field Q( √ 2) is studied in detail with a focus on representing Mersenne primes in the form x 2 + 7y 2 . It is also proved that x is divisible by 8 and y ≡ ±3 (mod 8) generalizing the result of F Lemmermeyer, first proved in [4] using Artin's Reciprocity law.