Search: id:A066408
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%I A066408
%S A066408 2,5,7,11,17,19,79,163,193,239,317,353,659,709,1049,1103,1759,2029,
%T A066408 5153,7541,9049,10453,23743,255361,534827
%N A066408 Eisenstein-Mersenne primes: numbers n such that the Eisenstein integer 
               (1-w)^n - 1 has prime norm, where w = - 1/2 + sqrt(-3)/2.
%C A066408 Analogue of Mersenne primes in Eisenstein integers.
%C A066408 The norm of a + b*w is (a+b*w)*(a+b*w^2).
%C A066408 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 A066408 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 (noe(AT)sspectra.com), Feb 02 2005
%D A066408 P. H. T. Beelen, Algebraic geometry and coding theory, Ph.D. Thesis, 
               Eindhoven, The Netherlands, September 2001.
%D A066408 J. M. Doumen, Ph.D. Thesis, Eindhoven, The Netherlands, to appear.
%D A066408 Mike Oakes, posting to primenumbers(AT)yahoogroups.com, Dec 24, 2001
%H A066408 C. Caldwell, <a href="http://primes.utm.edu/primes/download.php">The 
               largest known primes</a>
%H A066408 M. Oakes, <a href="http://groups.yahoo.com/group/primenumbers/message/
               4607">Eisenstein Mersenne and Fermat primes</a>
%H A066408 M. Oakes, <a href="http://www.mail-archive.com/mersenne@base.com/msg05162.html">
               A new series of Mersenne-like Gaussian primes</a>
%H A066408 M. Oakes, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0512&L=nmbrthry&T=0&F=&S=&P=2997">
               Posting to the Number Theory list</a>, Dec 27 2005
%e A066408 For n = 7, (1-w)^7 - 1 has norm 2269, a prime.
%e A066408 Or, for p=7, 3^7+3^4+1=2269, which is prime.
%Y A066408 The actual norms are in A066413. Cf. A000043, A057429.
%Y A066408 Sequence in context: A038611 A023213 A162575 this_sequence A142352 A062044 
               A077128
%Y A066408 Adjacent sequences: A066405 A066406 A066407 this_sequence A066409 A066410 
               A066411
%K A066408 nonn,nice,hard
%O A066408 1,1
%A A066408 Mike Oakes (mikeoakes2(AT)aol.com), Dec 24 2001

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