Re: question on class number for a certain n

On Wed, Nov 19, 2025 at 3:24 PM Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:

On Wed, Nov 19, 2025 at 10:16:28AM +0100, Bill Allombert wrote:
> 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.

I have found a formula (also for n>4 squarefree)
that only count positive increasing triples 0<=a<=b<=c.

The idea is to count 'exceptional' representations separately, that is the representations
that include repeated terms or the number 0, and use linear algebra.

? cnt(416666)
%2 = 339

Cheers,
Bill