A055029 Number of inequivalent Gaussian primes of norm n.

0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0

Offset

0,6

Comments

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-i).

References

R. K. Guy, Unsolved Problems in Number Theory, A16.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for Gaussian integers and primes

Formula

a(n) = A055028(n)/4.

a(n) = 2 if n is a prime = 1 (mod 4); a(n) = 1 if n is 2, or p^2 where p is a prime = 3 (mod 4); a(n) = 0 otherwise. - Franklin T. Adams-Watters, May 05 2006

a(n) = if n = 2 then 1 else 2*A079260(n) + A079261(A037213(n)). - Reinhard Zumkeller, Nov 11 2012

Example

There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i).

Mathematica

a[n_ /; PrimeQ[n] && Mod[n, 4] == 1] = 2; a[2] = 1; a[n_ /; (p = Sqrt[n]; PrimeQ[p] && Mod[p, 4] == 3)] = 1; a[_] = 0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 25 2011, after Franklin T. Adams-Watters *)

Prog

(Haskell)
a055029 2 = 1
a055029 n = 2 * a079260 n + a079261 (a037213 n)
-- Reinhard Zumkeller, Nov 11 2012
(PARI) a(n)=if(isprime(n), if(n%4==1, 2, n==2), if(issquare(n, &n) && isprime(n) && n%4==3, 1, 0)) \\ Charles R Greathouse IV, Feb 07 2017

Crossrefs

Cf. A055025, A055026, A055027, A055028.

Cf. A055664, A055665, A055666, A055667, A055668.

Sequence in context: A059483 A067618 A279255 * A126812 A008442 A338690

Adjacent sequences: A055026 A055027 A055028 * A055030 A055031 A055032

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jun 09 2000

Extensions

More terms from Reiner Martin, Jul 20 2001

Status

approved