A066413 - OEIS

%I #19 Sep 27 2022 13:07:24

%S 7,271,2269,176419,129159847,1162320517,

%T 49269609804781974450852068861184694669,

%U 589881151426658740854227725580736348850640632297373414091790995505756623268837

%N Eisenstein-Mersenne primes: primes of the form ((3 +/- sqrt(-3))/2)^p - 1.

%C Analogs of Mersenne primes in Eisenstein integers.

%C The norm of a + b*w is (a+b*w)*(a+b*w^2) = a^2 - a*b + b^2.

%D Mike Oakes, email dated Dec 24 2001 to primenumbers(AT)yahoogroups.com

%H K. Pershell and L. Huff, <a href="https://www.utm.edu/staff/caldwell/preprints/kpp/Paper2.pdf">Mersenne primes in imaginary quadratic number fields</a>, Apr 30 2002.

%e For n = 7, (1-w)^7 - 1 has norm 2269, a prime.

%t maxPi = 40; norm[p_] := 1 + 3^p - 2*3^(p/2)*Cos[p*Pi/6]; A066413 = {}; Do[ If[ PrimeQ[ np = norm[ Prime[k] ] ], AppendTo[ A066413, np] ], {k, 1, maxPi}]; A066413 (* _Jean-François Alcover_, Oct 16 2012 *)

%Y Cf. A066408.

%K nonn,nice

%O 1,1

%A _Mike Oakes_, Dec 24 2001

%E Name changed by _Arkadiusz Wesolowski_, Apr 27 2012