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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 18 21:20 EST 2020. Contains 332028 sequences. (Running on oeis4.)