

A103329


Numbers n such that (1+i)^n  i is a Gaussian prime.


3



0, 3, 4, 5, 8, 10, 16, 26, 29, 34, 73, 113, 122, 157, 178, 241, 353, 457, 997, 1042, 3041, 4562, 6434, 8506, 10141, 19378, 19882, 22426, 27529
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OFFSET

1,2


COMMENTS

Note that A027206 and A057429 treat Gaussian primes of a similar form. The remaining case, (1+i)^n + 1, is a Gaussian prime for n=1,2,3,4 only.
Let z = (1+i)^n  i. If z is not pure real or pure imaginary, then z is a Gaussian prime if the product of z and its conjugate is a rational prime. That product is 1 + 2^n  sin(n*Pi/4)*2^(1+n/2). z is real when n=1. z is imaginary when n=4k+2, in which case, z has magnitude 2^(2k+1)  (1)^k. These pure imaginary numbers are Gaussian primes when 2^(2k+1)1 is a Mersenne prime and 2k+1 = 3 (mod 4); that is, when n is twice an odd number in A112634.  T. D. Noe, Mar 07 2011


LINKS

Table of n, a(n) for n=1..29.


MATHEMATICA

fQ[n_] := PrimeQ[(1 + I)^n  I, GaussianIntegers > True]; Select[ Range[0, 30000], fQ]


CROSSREFS

Cf. A027206 ((1+i)^n + i is a Gaussian prime), A057429 ((1+i)^n  1 is a Gaussian prime).
Sequence in context: A079136 A235862 A228305 * A080726 A101210 A206445
Adjacent sequences: A103326 A103327 A103328 * A103330 A103331 A103332


KEYWORD

nonn


AUTHOR

T. D. Noe, Jan 31 2005


EXTENSIONS

a(25)a(29) from Robert G. Wilson v, Mar 02 2011.
0 prepended by T. D. Noe, Mar 07 2011


STATUS

approved



