Re: question on class number for a certain n

Function p3(n) in the attached code works for all n.

Regards,
Max



On Wed, Nov 19, 2025 at 3:24 PM Bill Allombert <
[email protected]> 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
>