A103329 Numbers n such that (1+i)^n - i is a Gaussian prime.

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

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 * A079347 A080726 A101210

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