The concept of Mersenne primes is studied in real quadratic fields with class number one. Computational results are given. The field Q(√ 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.