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PH2002

Perschell, Karaloine and Huff, Loran, "Mersenne primes in imaginary quadratic number fields," (2002) avaliable from http://www.utm.edu/staff/caldwell/preprints/kpp/Paper2.pdf.

Abstract: We examine all primes of the form bⁿ-1 in imaginary quadratic number fields, noting that besides a finite list of specific prime numbers, we find only three interesting classes of primes. Using the rational Mersenne primes (the first of the three cases) as a model, we follow Robert Spira and Mike Oakes in defining the Gaussian Mersenne primes to be the primes of the form (1±)^p-1 and the Eisenstein Mersenne primes to be the primes of the form ((3± sqrt(-3) /2)^p-1. We show how to characterize these via their norms, list the known examples and speculate on their distributions.