The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{ 2})$ 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 $y\equiv \pm3\pmod{ 8}$ generalizing the result of F Lemmermeyer, first proved in \cite{ LS} using Artin's Reciprocity law.
Categories: math.NT