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%I #91 Jun 21 2023 08:03:12
%S 2,5,7,11,17,19,79,163,193,239,317,353,659,709,1049,1103,1759,2029,
%T 5153,7541,9049,10453,23743,255361,534827,2237561,2888387,4043119
%N Numbers n such that the Eisenstein integer (1 - ω)^n - 1 has prime norm, where ω = -1/2 + sqrt(-3)/2.
%C Analog of Mersenne primes in Eisenstein integers.
%C The norm of a + b * ω is (a + b * ω) * (a + b * ω^2) = a^2 + a*b + b^2.
%C 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
%C 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
%C 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
%C Next term > 4400000. - _Serge Batalov_, Jun 20 2023
%D P. H. T. Beelen, Algebraic geometry and coding theory, Ph.D. Thesis, Eindhoven, The Netherlands, September 2001.
%D J. M. Doumen, Ph.D. Thesis, Eindhoven, The Netherlands, to appear.
%D Mike Oakes, posting to primenumbers(AT)yahoogroups.com, Dec 24 2001.
%H Pedro Berrizbeitia and Boris Iskra, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.189.311">Gaussian Mersenne and Eisenstein Mersenne primes</a>, Mathematics of Computation 79 (2010), pp. 1779-1791.
%H Chris Caldwell, <a href="https://t5k.org/primes/download.php">The largest known primes</a>
%H Chris Caldwell, <a href="https://t5k.org/top20/page.php?id=44">Generalized Unique primes</a>
%H Mike Oakes, <a href="http://groups.yahoo.com/group/primenumbers/message/4607">Eisenstein Mersenne and Fermat primes</a>
%H Mike Oakes, <a href="/A066408/a066408.txt">Eisenstein Mersenne and Fermat primes</a>, message 4607 in primenumbers Yahoo group, Dec 24, 2001.
%H Mike Oakes, <a href="http://www.mail-archive.com/mersenne@base.com/msg05162.html">A new series of Mersenne-like Gaussian primes</a>
%H Mike Oakes, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;186cb60a.0512">Posting to the Number Theory list</a>, Dec 27 2005.
%H K. Pershell and L. Huff, <a href="http://www.utm.edu/staff/caldwell/preprints/kpp/Paper2.pdf">Mersenne Primes in Imaginary Quadratic Number Fields</a>, (2002).
%H Eric Weissteins's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinInteger.html">Eisenstein Integer</a>
%e For n = 7, (1 - ω)^7 - 1 has norm 2269, a prime.
%e Or, for p = 7, 3^7 + 3^4 + 1 = 2269, which is prime.
%t 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 *)
%o (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
%Y The actual norms are in A066413.
%Y Cf. A000043, A010527, A057429, A125738, A125739.
%K nonn,nice,hard,more
%O 1,1
%A _Mike Oakes_, Dec 24 2001
%E a(26) from _Serge Batalov_, Mar 29 2014
%E a(27) from _Ryan Propper_ and _Serge Batalov_, Jun 18 2023
%E a(28) from _Ryan Propper_ and _Serge Batalov_, Jun 20 2023
%E Corrected link to NMBRTHRY posting. - _Serge Batalov_, Apr 01 2014
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