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

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, 15317227

Offset: 1

Author

Robert G. Wilson v, Sep 07 2000

Keywords

• nonn
• nice
• hard
• more

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

Examples

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

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, A new series of Mersenne-like Gaussian primes
• M. Oakes, Posting to the Number Theory list, Dec 27 2005
• K. Pershell and L. Huff, Mersenne Primes in Imaginary Quadratic Number Fields, (2002).
• Index entries for Gaussian integers and primes

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).

Programs

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 *)

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 */

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