Re: question on class number for a certain n

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Nov 18, 2025 at 10:59:53PM -0800, American Citizen wrote:
> Hi all:
> 
> I have been looking at representations of integers as the sum of 3 squares
> and things are very interesting.
> 
> A quote from Wolfram Math states
> 
> > The number of solutions of
> > 
> > (36)   x^2 + y^2 + z^2 = n
> > 
> > for a given n without restriction on the signs or relative sizes of x,
> > y, and z is given by r_3(n). Gauss proved that if n is squarefree and
> > n>4, then
> > 
> > (37)  r_3(n) =  24h(-n)  for n=3 (mod 8);
> >              =  12h(-4n) for n=1,2,5,6 (mod 8);
> >              =   0       for n=7 (mod 8)
> > 
> > (Arno 1992), where h(x) is the class number of x.
> subsequent post. Can r_3(n) be found?

This is a classical formula, but beware, it count all ordered triples (x,y,z)
in Z^3, so for example for n=17 there are 48 solutions instead of 2.

? 12*quadclassunit(-4*17).no
%1 = 48

and for 6844361

? 12*quadclassunit(-4*6844361).no
%2 = 69840

Cheers,
Bill.