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 A057429 Numbers n such that (1+i)^n - 1 times its conjugate is prime. 10

%I

%S 2,3,5,7,11,19,29,47,73,79,113,151,157,163,167,239,241,283,353,367,

%T 379,457,997,1367,3041,10141,14699,27529,49207,77291,85237,106693,

%U 160423,203789,364289,991961

%N Numbers n such that (1+i)^n - 1 times its conjugate is prime.

%C Equivalently, numbers n such that (1+i)^n - 1 is a Gaussian prime.

%C 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

%C 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]

%C 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

%C Primes p such that ((1+i)^p - 1)((1-i)^p - 1) is prime. Number 2 together with odd primes p such that the norm 2^p - (-1)^((p^2-1)/8)*2^((p+1)/2)+1 is prime. Note that Legendre symbol (2/p) = (-1)^((p^2-1)/8) as above. - _Thomas Ordowski_, Feb 20 2013

%D Mike Oakes, posting to the Mersenne list, Sep 07 2000.

%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 Marc Chamberland, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Binary BBP-Formulae for Logarithms...</a>, J. Integer Seqs., Vol. 6, 2003.

%H M. Oakes, <a href="http://www.mail-archive.com/mersenne@base.com/msg05162.html">A new series of Mersenne-like Gaussian primes</a>

%H M. Oakes, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0512&amp;L=nmbrthry&amp;T=0&amp;F=&amp;S=&amp;P=2997">Posting to the Number Theory list</a>, Dec 27 2005

%H <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>

%e Note that 4 is not in the sequence because (1+i)^4 - 1 = -5, which is an integer prime, but not a Gaussian prime.

%t Do[a = (1 + I)^n - 1; b = a*Conjugate[a]; If[PrimeQ[b], Print[n]], {n, 1, 160426}]

%o (PARI)

%o N=10^7; default(primelimit,N);

%o forprime(p=2,N,if(ispseudoprime(norm((1+I)^p-1)),print1(p,", ")));

%o /* _Joerg Arndt_, Jul 06 2011 */

%Y Cf. A000043, A066408, A007670, A007671, A027206.

%Y Cf. A027206 ((1+i)^n + i is a Gaussian prime), A103329 ((1+i)^n - i is a Gaussian prime).

%K nonn,nice,hard,more

%O 1,1

%A _Robert G. Wilson v_, Sep 07 2000

%E 364289 found by Nicholas Glover on Jun 02 2001 - Mike Oakes (mikeoakes2(AT)aol.com)

%E Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Aug 14 2002; revised by _N. J. A. Sloane_, Dec 28 2005

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