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

1,1

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

Table of n, a(n) for n=1..41.

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

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: A340418 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

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

Status

approved