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

TL;DR

This paper investigates the representation of Gaussian Mersenne primes in the form x^2+7y^2, revealing specific congruence properties and their splitting behavior in certain number field extensions, with generalizations and alternative proofs.

Contribution

It establishes new properties of Gaussian Mersenne primes related to their representation and splitting in number fields, extending previous results and providing alternative proofs using class field theory.

Findings

Gaussian Mersenne primes split completely in a cyclic quartic unramified extension of Q(√-14)

Such primes satisfy x ≡ ±1 mod 8 and y ≡ 0 mod 8 when representable as x^2+7y^2

Generalization to primes with d ≡ 7 mod 24

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.