A027206 Numbers m such that (1+i)^m + i is a Gaussian prime.

0, 1, 2, 3, 4, 6, 7, 8, 11, 14, 16, 19, 38, 47, 62, 79, 151, 163, 167, 214, 239, 254, 283, 367, 379, 1214, 1367, 2558, 4406, 8846, 14699, 49207, 77291, 160423, 172486, 221006, 432182, 1513678, 2515574

Offset: 1

Author

Ed Pegg Jr, Aug 07 2002

Keywords

• nonn

Comments

Equivalently, either (1+i)^m + i times its conjugate is an ordinary prime, or m == 2 (mod 4) and 2^(m/2) + (-1)^((m-2)/4) is an ordinary prime.

Let z = (1+i)^m + 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^m + sin(m*Pi/4)*2^(1+m/2). z is imaginary when m=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 == 1 (mod 4); that is, when m is twice an odd number in A112633. - T. D. Noe, Mar 07 2011

Links

• Index entries for Gaussian integers and primes

Crossrefs

Cf. A057429, A103329.

Programs

Mathematica

Select[Range[0,30000], PrimeQ[(1+I)^#+I, GaussianIntegers->True]&]

Extensions

More terms from Mike Oakes, Aug 07 2002

Edited by Dean Hickerson, Aug 14 2002

0 prepended by T. D. Noe, Mar 07 2011