正三角形的最小整数边长

葡萄糖

wayne 发表于 2020-9-11 17:37
如同mathe所说，四面体体积为0，得到\[\left|
\begin{array}{ccccc}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & a^2 & a^2 & d^2 \\
1 & a^2 & 0 & a^2 & e^2 \\
1 & a^2 & a^2 & 0 & f^2 \\
1 & d^2 & e^2 & f^2 & 0 \\
\end{array}
\right| =0\]
也就是 $a^4 -a^2(d^2 + e^2 + f^2) + (d^4 + e^4 + f^4-d^2 f^2 - e^2 f^2 - d^2 e^2) =0$,于是$a^2 = \frac{1}{2} \left(d^2+e^2+f^2 \pm \sqrt{3} \sqrt{2 d^2 e^2+2 d^2 f^2+2 e^2 f^2-d^4-e^4-f^4}\right)$

这个问题相当困难
Arnfried Kemnitz给出了如下不定方程的解
\[ \Large{a^4+x^4+y^4+z^4=x^2y^2+y^2z^2+x^2z^2+a^2 (x^2+y^2+z^2)} \]
\begin{align*}
\left\{\begin{split}
x&=\Big(u^2+v^2\Big)\Big(5u^2+8uv+5v^2\Big)\\
y&=7u^4+16u^3v+10u^2v^2+3v^4\\
z&=3u^4+10u^2v^2+16uv^3+7v^4\\
a&=8\Big(u-v\Big)\Big(u+v\Big)\Big(u^2+uv+v^2\Big)
\end{split}\right.
\end{align*}
参看《数论中未解决的问题》[加]盖伊（著）张明尧（译）
/2003-01-01 /科学出版社 P233 D19
https://mathworld.wolfram.com/RationalDistanceProblem.html
https://proofwiki.org/wiki/Small ... ances_from_Vertices

1. a^4 + x^4 + y^4 + z^4 - x^2 y^2 - x^2 z^2 - y^2 z^2 -
   
2.    a^2 (x^2 + y^2 + z^2) /. {
   
3.    x -> (u^2 + v^2) (5 u^2 + 8 u v + 5 v^2),
   
4.    y -> 7 u^4 + 16 u^3 v + 10 u^2 v^2 + 3 v^4,
   
5.    z -> 3 u^4 + 10 u^2 v^2 + 16 u v^3 + 7 v^4,
   
6.    a -> 8 (u - v) (u + v) (u^2 + u v + v^2)} // FullSimplify

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http://kuing.orzweb.net/viewthread.php?tid=7034&rpid=35459

葡萄糖

{{331},111,221,280}
{{331},49,285,296}

{{1729},211,1541,1560}
{{1729},824,915,1591}
{{1729},195,1544,1591}

{{1805},811,1029,1744}
{{1805},379,1464,1519}

{{2408},1023,1387,2045}
{{2408},715,1737,2353}

葡萄糖

葡萄糖 发表于 2020-9-14 09:25
这个问题相当困难
...

\begin{align*}
\left\{\begin{split}
a&=\dfrac{(2+\sqrt{3})^{2n+2}-(2-\sqrt{3})^{2n+2}}{\sqrt{3}}\\
x&=\dfrac{(2+\sqrt{3})^{2n+2}+(2-\sqrt{3})^{2n+2}+1}{3}\\
y&=\dfrac{(2+\sqrt{3})^{2n+2}+(2-\sqrt{3})^{2n+2}+1}{3}-
  \dfrac{(2+\sqrt{3})^{n+1}-(2-\sqrt{3})^{n+1}}{\sqrt{3}}\\
z&=\dfrac{(2+\sqrt{3})^{2n+2}+(2-\sqrt{3})^{2n+2}+1}{3}+
  \dfrac{(2+\sqrt{3})^{n+1}-(2-\sqrt{3})^{n+1}}{\sqrt{3}}
\end{split}\right.
\end{align*}

1. a^4 + x^4 + y^4 + z^4 - a^2 x^2 - a^2 y^2 - a^2 z^2 - x ^2 y^2 -
   
2.    x^2 z^2 -
   
3.    y^2 z^2 /. {a -> ((2 + Sqrt[3])^(2 n + 2) - (2 - Sqrt[3])^(
   
4.      2 n + 2))/Sqrt[3],
   
5.    x -> ((2 + Sqrt[3])^(2 n + 2) + (2 - Sqrt[3])^(2 n + 2) + 1)/3,
   
6.    y -> ((2 + Sqrt[3])^(2 n + 2) + (2 - Sqrt[3])^(2 n + 2) + 1)/
   
7.      3 - ((2 + Sqrt[3])^(n + 1) - (2 - Sqrt[3])^(n + 1))/Sqrt[3],
   
8.    z -> ((2 + Sqrt[3])^(2 n + 2) + (2 - Sqrt[3])^(2 n + 2) + 1)/
   
9.      3 + ((2 + Sqrt[3])^(n + 1) - (2 - Sqrt[3])^(n + 1))/Sqrt[
   
10.      3]} // FullSimplify

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