A066413 Eisenstein-Mersenne primes: primes of the form ((3 +/- sqrt(-3))/2)^p - 1.

7, 271, 2269, 176419, 129159847, 1162320517, 49269609804781974450852068861184694669, 589881151426658740854227725580736348850640632297373414091790995505756623268837

Offset

1,1

Comments

Analogs of Mersenne primes in Eisenstein integers.

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

References

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

Links

Table of n, a(n) for n=1..8.

K. Pershell and L. Huff, Mersenne primes in imaginary quadratic number fields, Apr 30 2002.

Example

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

Mathematica

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 *)

Crossrefs

Cf. A066408.

Sequence in context: A069449 A100465 A140031 * A332127 A222942 A289634

Adjacent sequences: A066410 A066411 A066412 * A066414 A066415 A066416

Keyword

nonn,nice

Author

Mike Oakes, Dec 24 2001

Extensions

Name changed by Arkadiusz Wesolowski, Apr 27 2012

Status

approved