Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #93 Aug 06 2024 09:20:00
%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="https://citeseerx.ist.psu.edu/pdf/571839efa5fa0561b70a184f0865cebf384f7c37">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