|
| |
|
|
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
|
Table of n, a(n) for n=1..40.
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
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 *)
|
|
|
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 * A065726 A215161 A227907
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
|
| |
|
|
