Gaussian Mersenne and Eisenstein Mersenne primes

The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas–Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth’s Theorem. Other properties of Gaussian Mersenne norms that contribute to the search for large primes are given. The Cubic Reciprocity Law is used to produce a primality test for Eisenstein Mersenne norms. The search for primes in both families (Gaussian Mersenne and Eisenstein Mersenne norms) was implemented in 2004 and ended in November 2005, when the largest primes, known at the time in each family, were found.