Mersenne primes in real quadratic fields

Abstract

The concept of Mersenne primes is studied in real quadratic fields with class number one. Computational results are given. The field ?(?2) is studied in detail with a focus on representing Mersenne primes in the form x2 + 7y2. It is also proved that x is divisible by 8 and y ? 3 (mod 8), generalizing a result of F. Lemmermeyer, first proved by H. W. Lenstra and P. Stevenhagen using Artin's reciprocity law.