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A057429
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Gaussian-Mersenne primes: numbers n such that (1+i)^n - 1 times its conjugate is prime.
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8
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2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Equivalently, numbers n such that (1+i)^n - 1 is a Gaussian prime.
Note that n must be a rational prime. Also note that (1+i)^n+i or (1+i)^n-i is also a Gaussian prime. - T. D. Noe, Jan 31 2005
Primes which are the norms of the Gaussian integers (1 + i)^n - 1 or (1 - i)^n - 1. [From Jonathan Vos Post, Feb 05 2010]
Let z = (1+i)^n - 1. The product of z and its conjugate is 1 + 2^n + cos(n*Pi/4)*2^(1+n/2). For n > 3, the primes are in A007670 or A007671 depending on whether n = {1,7} (mod 8) or n = {3,5} (mod 8), respectively. - T. D. Noe, Mar 07 2010
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REFERENCES
| Mike Oakes, posting to the Mersenne list, Sep 07 2000.
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LINKS
| Pedro Berrizbeitia and Boris Iskra, Gaussian Mersenne and Eisenstein Mersenne primes, Mathematics of Computation 79 (2010), pp. 1779-1791.
C. Caldwell, The largest known primes
Marc Chamberland, Binary BBP-Formulae for Logarithms..., J. Integer Seqs., Vol. 6, 2003.
M. Oakes, A new series of Mersenne-like Gaussian primes
M. Oakes, Posting to the Number Theory list, Dec 27 2005
Index entries for Gaussian integers and primes
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EXAMPLE
| Note that 4 is not in the sequence because (1+i)^4 - 1 = -5, which is an integer prime, but not a Gaussian prime.
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MATHEMATICA
| Do[a = (1 + I)^n - 1; b = a*Conjugate[a]; If[PrimeQ[b], Print[n]], {n, 1, 160426}]
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PROG
| (PARI)
N=10^7; default(primelimit, N);
forprime(p=2, N, if(ispseudoprime(norm((1+I)^p-1)), print1(p, ", ")));
/* Joerg Arndt, Jul 06 2011 */
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CROSSREFS
| Cf. A000043, A066408, A007670, A007671, A027206.
Cf. A027206 ((1+i)^n + i is a Gaussian prime), A103329 ((1+i)^n - i is a Gaussian prime).
Sequence in context: A039726 A115617 A003064 * A065726 A118985 A092728
Adjacent sequences: A057426 A057427 A057428 * A057430 A057431 A057432
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KEYWORD
| nonn,nice,hard
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 07 2000
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EXTENSIONS
| 364289 found by Nicholas Glover on Jun 02 2001 - Mike Oakes (mikeoakes2(AT)aol.com)
Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Aug 14 2002; revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 28 2005
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