Re: question on class number for a certain n

Max:

I checked using your p3(n) function, and it indeed matches my results 
for the first 237 relative maximums as N gradually climbs in value.

Thank you for posting this

Randall

On 11/19/25 14:37, Max Alekseyev wrote:
> 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
>