

A066408


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


9



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
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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


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



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



KEYWORD

nonn,nice,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



