Gaussian Mersenne Primes of the form $x^2+dy^2$

Abstract

In this paper we study Gaussian ring $\Z[i]$ with a focus on representing Gaussian Mersenne primes $G_p$ in the form $x^2+7y^2$. Interestingly when such a form exists, one can observe that, $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$. To prove this property of Gaussian Mersenne primes, we show that Gaussian Mersenne primes splits completely in the cyclic quartic unramified extension of $\Q(\sqrt{-14})$ and have a trivial Artin symbol in this extension. We generalize this result for $d\equiv 7\pmod{24}$. We also attempt to give an alternate proof using Artin's reciprocity law, which was earlier given by H. W. Lenstra and P. Stevenhagen to prove a similar property on ordinary Mersenne Primes.