A066408 Numbers n such that the Eisenstein integer (1 - ω)^n - 1 has prime norm, where ω = -1/2 + sqrt(-3)/2.

2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, 2888387, 4043119

Offset

1,1

Comments

Analog of Mersenne primes in Eisenstein integers.

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

Indices for which the Eisenstein-Mersenne numbers are primes. The p-th Eisenstein-Mersenne number can be written as 3^p - Legendre(3, p) * 3^((p + 1)/2) + 1. Note the enormous gap between 23743 and 255361. A modified version of Chris Nash's PFGW program was used to find the last term. - Jeroen Doumen (doumen(AT)win.tue.nl), Oct 31 2002

Let q be the integer quaternion (3 + i + j + k)/2. Then q^n - 1 is a quaternion prime for these n; that is, the norm of q^n - 1 is a rational prime. - T. D. Noe, Feb 02 2005

The actual norms also belong to the class of Generalized Unique primes (see Links section), that is primes which have a period of expansion of 1/p (in some general, non-decimal system) that it shares with no other prime. - Serge Batalov, Mar 29 2014

Next term > 4400000. - Serge Batalov, Jun 20 2023

References

P. H. T. Beelen, Algebraic geometry and coding theory, Ph.D. Thesis, Eindhoven, The Netherlands, September 2001.

J. M. Doumen, Ph.D. Thesis, Eindhoven, The Netherlands, to appear.

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

Links

Table of n, a(n) for n=1..28.

Pedro Berrizbeitia and Boris Iskra, Gaussian Mersenne and Eisenstein Mersenne primes, Mathematics of Computation 79 (2010), pp. 1779-1791.

Chris Caldwell, The largest known primes

Chris Caldwell, Generalized Unique primes

Mike Oakes, Eisenstein Mersenne and Fermat primes

Mike Oakes, Eisenstein Mersenne and Fermat primes, message 4607 in primenumbers Yahoo group, Dec 24, 2001.

Mike Oakes, A new series of Mersenne-like Gaussian primes

Mike Oakes, Posting to the Number Theory list, Dec 27 2005.

K. Pershell and L. Huff, Mersenne Primes in Imaginary Quadratic Number Fields, (2002).

Eric Weissteins's World of Mathematics, Eisenstein Integer

Example

For n = 7, (1 - ω)^7 - 1 has norm 2269, a prime.
Or, for p = 7, 3^7 + 3^4 + 1 = 2269, which is prime.

Mathematica

maxPi = 3000; primeNormQ[p_] := PrimeQ[1 + 3^p - 2*3^(p/2)*Cos[(p*Pi)/6]]; A066408 = {}; Do[ If[primeNormQ[p = Prime[k]], Print[p]; AppendTo[A066408, p]], {k, 1, maxPi}]; A066408 (* Jean-François Alcover, Oct 21 2011 *)

Prog

(PARI) print1("2, "); /*the only even member; it is special*/ forprime(n=3, 2029, if(ispseudoprime(3^n-kronecker(3, n)*3^((n+1)/2)+1), print1(n, ", "))) \\ Serge Batalov, Mar 29 2014

Crossrefs

The actual norms are in A066413.

Cf. A000043, A010527, A057429, A125738, A125739.

Sequence in context: A265761 A023213 A162575 * A142352 A290012 A062044

Adjacent sequences: A066405 A066406 A066407 * A066409 A066410 A066411

Keyword

nonn,nice,hard,more

Author

Mike Oakes, Dec 24 2001

Extensions

a(26) from Serge Batalov, Mar 29 2014

a(27) from Ryan Propper and Serge Batalov, Jun 18 2023

a(28) from Ryan Propper and Serge Batalov, Jun 20 2023

Corrected link to NMBRTHRY posting. - Serge Batalov, Apr 01 2014

Status

approved