Re: question on class number for a certain n

On 2025-11-19 07:59, American Citizen wrote:
> Hi all:
> 
> I have been looking at representations of integers as the sum of 3
> squares and things are very interesting.
> 
If you want to visualize all sum of 3 squares, I created two 
applications for that:
https://stamm-wilbrandt.de/en/#long-term_objective

Regards,

Hermann.

> 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.
>> 
>> The generating function for r_3(n) is given by
>> 
>> (38)  sum_(n=0)^(infty)r_3(n)x^n = theta_3^3(x)
>> (39)   = 1+6x+12x^2+8x^3+6x^4+24x^5+24x^6+12x^8+30x^9+...
>> 
>> Example:
>> 
>>    n = 6844361
>> 
>> Class number h(-4n) is what ???
> There are a lot of triads for this number n. I will share what I found
> in a subsequent post. Can r_3(n) be found?
> 
> Randall