求方程32(x^2+y^2+z^2)=17(x+y+z)^2的本原解

northwolves

葡萄糖 发表于 2023-1-13 15:34
不知道如下朴实无华的解是不是所有解
\begin{align*}
\left\{

这个应该是其中的一组解吧

northwolves

设\begin{align*}
\left\{
\begin{split}
x&=r_1t^2+r_2t+r_3\\
y&=r_4t^2+r_5t+r_6\\
z&=r_7t^2+r_8t+r_9
\end{split}
\right.
\end{align*}

$t^4 (15 r_1^2 + (-34 r_4 - 34 r_7) r_1 + 15 r_4^2 + 15 r_7^2 - 34 r_4 r_7) + 2 t^3  (-17 r_2 (r_4 + r_7) + r_4 (15 r_5 - 17 r_8) - r_7 (17 r_5 - 15 r_8) + r_1 (15 r_2 - 17 (r_5 + r_8))) + t^2 (15 r_2^2 + (-34 r_5 - 34 r_8) r_2 + 15 r_5^2 + 15 r_8^2 - 34 r_3 (r_4 + r_7) - 34 r_5 r_8 + 2 r_4 (15 r_6 - 17 r_9) + r_7 (30 r_9 - 34 r_6) + 2 r_1 (15 r_3 - 17 (r_6 + r_9))) + 2 t (-17 r_3 (r_5 + r_8) + r_5 (15 r_6 - 17 r_9) - r_8 (17 r_6 - 15 r_9) + r_2 (15 r_3 - 17 (r_6 + r_9))) +  (15 r_3^2 + (-34 r_6 - 34 r_9) r_3 + 15 r_6^2 + 15 r_9^2 - 34 r_6 r_9)=0$

\begin{align*}
\left\{
\begin{split}
15 r_1^2 + (-34 r_4 - 34 r_7) r_1 + 15 r_4^2 + 15 r_7^2 - 34 r_4 r_7=&0\\
-17 r_2 (r_4 + r_7) + r_4 (15 r_5 - 17 r_8) - r_7 (17 r_5 - 15 r_8) + r_1 (15 r_2 - 17 (r_5 + r_8))=&0\\
15 r_2^2 + (-34 r_5 - 34 r_8) r_2 + 15 r_5^2 + 15 r_8^2 - 34 r_3 (r_4 + r_7) - 34 r_5 r_8 + 2 r_4 (15 r_6 - 17 r_9) + r_7 (30 r_9 - 34 r_6) + 2 r_1 (15 r_3 - 17 (r_6 + r_9))=&0\\
-17 r_3 (r_5 + r_8) + r_5 (15 r_6 - 17 r_9) - r_8 (17 r_6 - 15 r_9) + r_2 (15 r_3 - 17 (r_6 + r_9))=&0\\
15 r_3^2 + (-34 r_6 - 34 r_9) r_3 + 15 r_6^2 + 15 r_9^2 - 34 r_6 r_9=&0
\end{split}
\right.
\end{align*}

northwolves

待定系数法5个方程解出9个未知数，理论上应该有很多解

northwolves

\begin{align*}
\left\{
\begin{split}
15(r_1^2+r_4^2+r_7^2)-34(r_1r_4+r_1r_7+r_4r_7)=0\\
15(r_2^2+r_5^2+r_8^2)-34(r_2r_5+r_2r_8+r_5r_8)=0\\
15(r_3^2+r_6^2+r_9^2)-34(r_3r_6+r_3r_9+r_6r_9)=0\\
15(r_1r_2+r_4r_5+r_7r_8)-17(r_2r_4+r_2r_7+r_4r_8+r_5r_7+r_1r_5+r_1r_8)=0\\
15(r_1r_3+r_4r_6+r_7r_9)-17(r_3r_4+r_3r_7+r_4r_9+r_6r_7+r_1r_6+r_1r_9)=0\\
15(r_2r_3+r_5r_6+r_8r_9)-17(r_3r_5+r_3r_8+r_5r_9+r_6r_8+r_2r_6+r_2r_9)=0\\
\end{split}
\right.
\end{align*}

葡萄糖

感觉下面这组解可以涵盖所有整数解
\begin{align*}
\left\{
\begin{split}
x&=8 (21 u + 3 v - 5 w) (9 u + 5 v - 3 w)\\
y&=(45 u + 77 v - 27 w) (3 u + 19 v - 5 w)\\
z&=(15 u - 9 v - w) (39 u - 65 v + 7 w)
\end{split}
\right.
\end{align*}

1. 32 (x^2 + y^2 + z^2) - 17 (x + y + z)^2 /. {x -> 8 (21 u + 3 v - 5 w) (9 u + 5 v - 3 w),
   
2.    y -> (45 u + 77 v - 27 w) (3 u + 19 v - 5 w),
   
3.    z -> (15 u - 9 v - w) (39 u - 65 v + 7 w)} // Factor

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葡萄糖

类似的三元丢番图方程
https://mathhelpplanet.com/viewtopic.php?f=48&t=77132
https://euler.jakumo.org/problems/view/785.html