

A066408


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


8



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
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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 EisensteinMersenne numbers are primes. The pth EisensteinMersenne 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, nondecimal system) that it shares with no other prime.  Serge Batalov, Mar 29 2014
Next term > 2300000.  Serge Batalov, Mar 29 2014


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..26.
Pedro Berrizbeitia and Boris Iskra, Gaussian Mersenne and Eisenstein Mersenne primes, Mathematics of Computation 79 (2010), pp. 17791791.
Chris Caldwell, The largest known primes
Chris Caldwell, Generalized Unique primes
Mike Oakes, Eisenstein Mersenne and Fermat primes
Mike Oakes, A new series of Mersennelike 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 (* JeanFrançois Alcover, Oct 21 2011 *)


PROG

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


CROSSREFS

The actual norms are in A066413.
Cf. A000043, A057429.
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
Corrected link to NMBRTHRY posting.  Serge Batalov, Apr 01 2014


STATUS

approved



