cp's OEIS Frontend

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

%I A103329 #12 Jul 14 2012 13:52:00
%S A103329 0,3,4,5,8,10,16,26,29,34,73,113,122,157,178,241,353,457,997,1042,
%T A103329 3041,4562,6434,8506,10141,19378,19882,22426,27529
%N A103329 Numbers n such that (1+i)^n - i is a Gaussian prime.
%C A103329 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.
%C A103329 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
%t A103329 fQ[n_] := PrimeQ[(1 + I)^n - I, GaussianIntegers -> True]; Select[ Range[0, 30000], fQ]
%Y A103329 Cf. A027206 ((1+i)^n + i is a Gaussian prime), A057429 ((1+i)^n - 1 is a Gaussian prime).
%K A103329 nonn
%O A103329 1,2
%A A103329 _T. D. Noe_, Jan 31 2005
%E A103329 a(25)-a(29) from _Robert G. Wilson v_, Mar 02 2011.
%E A103329 0 prepended by _T. D. Noe_, Mar 07 2011