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
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
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
a(41) from Ryan Propper and Serge Batalov, Jun 20 2023
STATUS
approved