A055028 Number of Gaussian primes of norm n.

0, 0, 4, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane, Jun 09 2000

Keywords

• nonn
• easy
• nice

Comments

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

Examples

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

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A055025-A055029, A055664-... , A295996.

Programs

Maple

A055028 := proc(n::integer)
    local c,a,b ;
    c := 0 ;
    for a from -n to n do
        if issqr(n-a^2) then
            b := sqrt(n-a^2) ;
            if GaussInt[GIprime](a+b*I) and a^2+b^2=n then
                if b = 0 then
                    c := c+1 ; # a+i*b and a-i*b
                else
                    c := c+2 ; # a+i*b and a-i*b
                end if;
            end if;
        end if;
    end do:
    c ;
end proc:
seq( A055028(n),n=0..50) ; # R. J. Mathar, Jul 22 2021

Mathematica

a[n_ /; PrimeQ[n] && Mod[n, 4] == 1] = 8; a[2] = 4; a[n_ /; (p = Sqrt[n]; PrimeQ[p] && Mod[p, 4] == 3)] = 4; a[] = 0; Table[ a[n], {n, 0, 100}] (* _Jean-François Alcover, Jul 30 2013, after Franklin T. Adams-Watters *)

Formula

a(n) = 4 * A055029(n). - Franklin T. Adams-Watters, May 05 2006

Extensions

More terms from Reiner Martin, Jul 20 2001