Timeline for How to prove this problem about ternary quadratic form?
Current License: CC BY-SA 4.0
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| Sep 17 at 14:56 | comment | added | GH from MO | In my previous comment, $ac-b^2=n$ should be replaced by $b^2-4ac=-4n$. | |
| Sep 17 at 14:56 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 16 at 23:20 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 16 at 21:49 | comment | added | GH from MO | @IosifPinelis I looked up Dickson's book "History of the theory of numbers". He mentions on page 265 of volume 2, and also on page 109 of volume 3, that Kronecker (1860) used elliptic functions to prove Gauss's result in the form $r_3(n)=24F(n)-12G(n)$. Here $G(n)$ is as above, and $F(n)$ is the weighted number of classes of $[a,b,c]$ with $b$ even, $a$ or $c$ odd, $ac-b^2=n$. He also references the result that $F(n)=G(n)$ for $n\equiv 1,2\pmod{4}$, hence in fact $r_3(n)=12G(n)$ for such $n$. | |
| Sep 16 at 20:08 | comment | added | GH from MO | @IosifPinelis I cannot give a reference off-hand. I guess it is in Disquisitiones. The usual formulation relates primitive lattice points to classes of primitive forms. As the two sides transform in the same way, the conversion is straightforward. Moreover, these days the condition $2\mid b$ is dropped for the forms, hence a slightly different class number is needed. The two kinds of class numbers are related by $H(-n)=h(-4n)$, and my memory tells that this one is in Dickson. There is a treatment with $h(-4n)$ as opposed to $H(-n)$ in Grosswald: Representations of integers as sums of squares. | |
| Sep 16 at 20:00 | comment | added | Iosif Pinelis | Can you give a detailed reference to Gauss (1801)? Is there an accessible English translation of that proof? | |
| Sep 16 at 19:05 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 16 at 18:49 | vote | accept | 8451543498 | ||
| Sep 16 at 18:46 | history | answered | GH from MO | CC BY-SA 4.0 |