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%I #86 Jun 21 2023 06:34:46
%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 160423,203789,364289,991961,1203793,1667321,3704053,4792057,15317227
%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. - _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
%C The exhaustive search for all a(n)<5000000 is now complete. - _Serge Batalov_, Sep 06 2014
%C The primes generated by these series are also generalized unique primes. They can be represented as Phi(4, 2^((p+1)/2) - (2/p))/2, where (2/p) is the Legendre symbol (Cf. link to Generalized unique primes page at UTM). - _Serge Batalov_, Sep 08 2014
%D Mike Oakes, posting to the Mersenne list, Sep 07 2000.
%H Pedro Berrizbeitia and Boris Iskra, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02324-0">Gaussian Mersenne and Eisenstein Mersenne primes</a>, Mathematics of Computation 79 (2010), pp. 1779-1791.
%H C. Caldwell, <a href="https://t5k.org/primes/download.php">The largest known primes</a>
%H C. Caldwell, <a href="https://t5k.org/top20/page.php?id=44">Generalized unique primes</a>
%H Marc Chamberland, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Chamberland/chamberland60.html">Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes</a>, J. Integer Seqs., Vol. 6, 2003.
%H MersenneForum, <a href="http://mersenneforum.org/showthread.php?t=19235">Gaussian Mersenne norm project coordination</a>
%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="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 <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}] (* Wilson *)
%t Select[Range[1000], PrimeQ[((1 + I)^# - 1)Conjugate[(1 + I)^# - 1]] &] (* _Alonso del Arte_, May 01 2014 *)
%t Select[Range[48*10^5],PrimeQ[(1+I)^#-1,GaussianIntegers->True]&] (* _Harvey P. Dale_, Dec 30 2018 *)
%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_
%E Edited by _Dean Hickerson_, Aug 14 2002; revised by _N. J. A. Sloane_, Dec 28 2005
%E a(37)-a(38) from B. Jaworski (found in 2006 and 2011) - _Serge Batalov_, May 01 2014
%E a(39)-a(40) from _Serge Batalov_, Sep 06 2014
%E a(41) from _Ryan Propper_ and _Serge Batalov_, Jun 20 2023
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