Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
As is the case (analogous) for the sum of three squares, for the number $1$ we do not get the same ratios as for other numbers $1 \pmod {72},$ those being together in lines with the output N 1 in the column for %72
As is the case (analogous) for the sum of three squares, for the number $1$ we do not get the same ratios as for other numbers $1 \pmod {72},$ those being together in lines with the output N 1 in the column for %72
Made a little summary: the ratios $(r_0/h(4M), r_1/h(4M))$$(r_0/h(-4M), r_1/h(-4M))$ are split three ways when $\gcd(M,6) = 1.$ We get ratios $(12,4)$ when $M \equiv 1, 13,25,37,49,61 \pmod {72},$ which is all $M \equiv 1 \pmod {12}.$ We get ratios $(24,16)$ when $M \equiv 7,31,55 \pmod {72},$ which is all $M \equiv 7 \pmod {24}.$We get ratios $(16,8)$ when $M \equiv 19,43,67 \pmod {72},$ which is all $M \equiv 19 \pmod {24}.$
Made a little summary: the ratios $(r_0/h(4M), r_1/h(4M))$ are split three ways when $\gcd(M,6) = 1.$ We get ratios $(12,4)$ when $M \equiv 1, 13,25,37,49,61 \pmod {72},$ which is all $M \equiv 1 \pmod {12}.$ We get ratios $(24,16)$ when $M \equiv 7,31,55 \pmod {72},$ which is all $M \equiv 7 \pmod {24}.$We get ratios $(16,8)$ when $M \equiv 19,43,67 \pmod {72},$ which is all $M \equiv 19 \pmod {24}.$
Made a little summary: the ratios $(r_0/h(-4M), r_1/h(-4M))$ are split three ways when $\gcd(M,6) = 1.$ We get ratios $(12,4)$ when $M \equiv 1, 13,25,37,49,61 \pmod {72},$ which is all $M \equiv 1 \pmod {12}.$ We get ratios $(24,16)$ when $M \equiv 7,31,55 \pmod {72},$ which is all $M \equiv 7 \pmod {24}.$We get ratios $(16,8)$ when $M \equiv 19,43,67 \pmod {72},$ which is all $M \equiv 19 \pmod {24}.$
You have a good answer; hereHere are some computations about primitive representations by two forms, $g_0$ is the underlying $x^2 + y^2 + 3 z^2 + xy,$ while $g_1$ is your $x^2 + 3 y^2 + 3 z^2.$ I was a little surprised to find a distinction requiring mod 72, mod 36 was not enough. Compare $7 \pmod {72}$ to $43 \pmod {72}.$ Alright, it amounts to $\pmod {24}.$ When I have time I will compute the cases divisible by 2 or 3 or both.
You have a good answer; here are some computations about primitive representations by two forms, $g_0$ is the underlying $x^2 + y^2 + 3 z^2 + xy,$ while $g_1$ is your $x^2 + 3 y^2 + 3 z^2.$ I was a little surprised to find a distinction requiring mod 72, mod 36 was not enough. Compare $7 \pmod {72}$ to $43 \pmod {72}.$ Alright, it amounts to $\pmod {24}.$ When I have time I will compute the cases divisible by 2 or 3 or both.
Here are some computations about primitive representations by two forms, $g_0$ is the underlying $x^2 + y^2 + 3 z^2 + xy,$ while $g_1$ is your $x^2 + 3 y^2 + 3 z^2.$ I was a little surprised to find a distinction requiring mod 72, mod 36 was not enough. Compare $7 \pmod {72}$ to $43 \pmod {72}.$ Alright, it amounts to $\pmod {24}.$ When I have time I will compute the cases divisible by 2 or 3 or both.
