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

Author

Mike Oakes, Dec 24 2001

Keywords

• nonn
• nice
• hard
• more

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

Examples

			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.
		

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

• 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

Crossrefs

The actual norms are in A066413.

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

Programs

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 *)

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

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