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Search: id:A066408
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| A066408 |
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Eisenstein-Mersenne primes: numbers n such that the Eisenstein integer (1-w)^n - 1 has prime norm, where w = - 1/2 + sqrt(-3)/2. |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Analogue of Mersenne primes in Eisenstein integers.
The norm of a + b*w is (a+b*w)*(a+b*w^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 (noe(AT)sspectra.com), Feb 02 2005
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REFERENCES
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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
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LINKS
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C. Caldwell, The largest known primes
M. Oakes, Eisenstein Mersenne and Fermat primes
M. Oakes, A new series of Mersenne-like Gaussian primes
M. Oakes, Posting to the Number Theory list, Dec 27 2005
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EXAMPLE
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For n = 7, (1-w)^7 - 1 has norm 2269, a prime.
Or, for p=7, 3^7+3^4+1=2269, which is prime.
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CROSSREFS
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The actual norms are in A066413. Cf. A000043, A057429.
Sequence in context: A038611 A023213 A162575 this_sequence A142352 A062044 A077128
Adjacent sequences: A066405 A066406 A066407 this_sequence A066409 A066410 A066411
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KEYWORD
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nonn,nice,hard
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AUTHOR
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Mike Oakes (mikeoakes2(AT)aol.com), Dec 24 2001
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