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 A057429 Numbers n such that (1+i)^n - 1 times its conjugate is prime. 10
 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, 1203793, 1667321, 3704053, 4792057 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. - 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 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 I have found that 4792057 is a term in the sequence but I can't yet confirm its position in the sequence. - Serge Batalov, Apr 05 2014 The exhaustive search for all a(n)<5000000 is now complete. - Serge Batalov, Sep 06 2014 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 REFERENCES Mike Oakes, posting to the Mersenne list, Sep 07 2000. 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 C. Caldwell, Generalized unique primes Marc Chamberland, Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes, J. Integer Seqs., Vol. 6, 2003. MersenneForum, Gaussian Mersenne norm project coordination M. Oakes, Posting to the Number Theory list, Dec 27 2005 K. Pershell and L. Huff, Mersenne Primes in Imaginary Quadratic Number Fields, (2002). 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. MATHEMATICA Do[a = (1 + I)^n - 1; b = a * Conjugate[a]; If[PrimeQ[b], Print[n]], {n, 1, 160426}] (* Wilson *) Select[Range[1000], PrimeQ[((1 + I)^# - 1)Conjugate[(1 + I)^# - 1]] &] (* Alonso del Arte, May 01 2014 *) Select[Range[48*10^5], PrimeQ[(1+I)^#-1, GaussianIntegers->True]&] (* Harvey P. Dale, Dec 30 2018 *) 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 */ 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 * A137814 A065726 A215161 Adjacent sequences:  A057426 A057427 A057428 * A057430 A057431 A057432 KEYWORD nonn,nice,hard,more AUTHOR Robert G. Wilson v, Sep 07 2000 EXTENSIONS 364289 found by Nicholas Glover on Jun 02 2001 - Mike Oakes Edited by Dean Hickerson, Aug 14 2002; revised by N. J. A. Sloane, Dec 28 2005 a(37)-a(38) from B. Jaworski (found in 2006 and 2011) - Serge Batalov, May 01 2014 a(39)-a(40) from Serge Batalov, Sep 06 2014 STATUS approved

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Last modified January 22 11:56 EST 2019. Contains 319363 sequences. (Running on oeis4.)