Gaussian Mersenne Primes of the form $x^2+d{y}^2$

Abstract

In this paper we study Gaussian ring $\mathbb{Z}[i]$ 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, $x\equiv \pm 1(\mathrm{mod}8)$ and $y\equiv 0(\mathrm{mod}8)$. 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 $d\equiv 7(\mathrm{mod}24)$. 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.

arXiv comment11 Pages

References in corpus (1)

• Mersenne Primes in Real Quadratic Fields

Cited by in corpus (1)

• Primes of the form $x^2+d{y}^2$ with $x\equiv 0(\mathrm{mod}N)$ or $y\equiv 0(\mathrm{mod}N)$