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

0,3

Comments

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

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

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A255328 A321433 A016678 * A330316 A028591 A048728

Adjacent sequences: A055025 A055026 A055027 * A055029 A055030 A055031

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jun 09 2000

Extensions

More terms from Reiner Martin, Jul 20 2001

Status

approved