标题： 梅森素数在实二次域中的研究 显示英文标题

标题： Mersenne Primes in Real Quadratic Fields

摘要： 研究了类数为 1 的实二次域中的梅森素数概念。 给出了计算结果。 详细研究了域$Q(\sqrt{2})$，并重点讨论了以形式$x^{2}+7y^{2}$表示梅森素数的问题。 还证明了$x$可被 8 整除以及$y\equiv \pm3\pmod{8}$推广了 F Lemmermeyer 的结果，该结果首先由 Artin 的互反律在\cite{LS}中证明。 显示英文摘要

摘要： 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.