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# [原创] 正三角形的最小整数边长

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 化简后好像要求 $(-d^4 -e^4 -f^4 + 2e^2 d^2 + 2f^2 d^2 + 2f^2 e^2 ) a^2 = f^2 e^2 d^2$ 所以要求$-d^4 -e^4 -f^4 + 2e^2 d^2 + 2f^2 d^2 + 2f^2 e^2$是完全平方，而且$a|f e d$

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 如同mathe所说，四面体体积为0，得到$\begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & a^2 & a^2 & d^2 \\ 1 & a^2 & 0 & a^2 & e^2 \\ 1 & a^2 & a^2 & 0 & f^2 \\ 1 & d^2 & e^2 & f^2 & 0 \\ \end{vmatrix} =0$ 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$a^2 = \frac{1}{2} \left(d^2+e^2+f^2 \pm \sqrt{3} \sqrt{2 d^2 e^2+2 d^2 f^2+2 e^2 f^2-d^4-e^4-f^4}\right)$ 暴力美学，算得最小的解是 ${5, 5,5, 3, 7, 8}, {5, 5,5, 16, 19, 21}$ {5,{{3,5,7,8},{5,16,19,21}}} {7,{{3,5,7,8},{7,8,13,15},{7,33,37,40}}} {8,{{3,5,7,8},{7,8,13,15}}} {9,{{9,15,21,24},{9,56,61,65}}} {10,{{6,10,14,16},{10,32,38,42}}} {11,{{11,24,31,35},{11,85,91,96}}} {13,{{7,8,13,15},{13,35,43,48}}} {14,{{6,10,14,16},{14,16,26,30},{14,66,74,80}}} {15,{{7,8,13,15},{9,15,21,24},{15,25,35,40},{15,48,57,63}}} {16,{{5,16,19,21},{6,10,14,16},{14,16,26,30},{16,39,49,55}}} {19,{{5,16,19,21},{19,80,91,99}}} {20,{{12,20,28,32},{20,64,76,84}}} {21,{{5,16,19,21},{9,15,21,24},{21,24,39,45},{21,35,49,56}}} {24,{{9,15,21,24},{11,24,31,35},{21,24,39,45},{24,40,56,64}}} {26,{{14,16,26,30},{26,70,86,96}}} {28,{{12,20,28,32},{28,32,52,60}}} {30,{{14,16,26,30},{18,30,42,48},{30,50,70,80}}} {32,{{10,32,38,42},{12,20,28,32},{28,32,52,60},{32,45,67,77}}} {33,{{7,33,37,40},{33,55,77,88}}} {35,{{11,24,31,35},{13,35,43,48},{15,25,35,40},{21,35,49,56},{35,40,65,75}}} {39,{{16,39,49,55},{21,24,39,45}}} {40,{{7,33,37,40},{15,25,35,40},{24,40,56,64},{35,40,65,75},{40,51,79,91}}} {42,{{10,32,38,42},{18,30,42,48},{42,48,78,90}}} {45,{{21,24,39,45},{27,45,63,72},{32,45,67,77}}} {48,{{13,35,43,48},{15,48,57,63},{18,30,42,48},{22,48,62,70},{42,48,78,90}}} {49,{{16,39,49,55},{21,35,49,56}}} {55,{{16,39,49,55},{33,55,77,88}}} {56,{{9,56,61,65},{21,35,49,56},{24,40,56,64}}} {60,{{28,32,52,60},{36,60,84,96}}} {63,{{15,48,57,63},{17,63,73,80},{27,45,63,72}}} {64,{{20,64,76,84},{24,40,56,64}}} {65,{{9,56,61,65},{35,40,65,75}}} {70,{{22,48,62,70},{26,70,86,96},{30,50,70,80}}} {77,{{32,45,67,77},{33,55,77,88}}} {80,{{14,66,74,80},{17,63,73,80},{19,80,91,99},{30,50,70,80}}} {84,{{20,64,76,84},{36,60,84,96}}} {91,{{11,85,91,96},{19,80,91,99},{40,51,79,91}}} {96,{{11,85,91,96},{26,70,86,96},{36,60,84,96}}} 复制代码

楼主| 发表于 2020-9-11 18:33:31 | 显示全部楼层
 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 \\ 要求是内点,不能在边上或者外部。

 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 \\ 从中提出合乎内点要求的结果。 {{{7, 33, 37}, {15, 25, 35}}, 40} {{{13, 35, 43}, {18, 30, 42}}, 48} {{{14, 66, 74}, {17, 63, 73}}, 80} {{{11, 85, 91}, {26, 70, 86}, {36, 60, 84}}, 96}复制代码

### 点评

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

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 f[n_]:=Times@@Cases[FactorInteger[n],{x_,y_}->x^Floor[y/2]] Do[g=Select[Divisors[f[a^4-a^2b^2+b^4-a^2c^2-b^2c^2+c^4]],(a^4-a^2b^2+b^4-a^2c^2-b^2c^2+c^4-a^2#^2-b^2#^2-c^2#^2+#^4==0&&c<#

### 点评

 直接用公式在a,b,c较大的时候效率更高些： Do[d=Round[Sqrt[a^2+b^2+c^2+Sqrt[3]Sqrt[-a^4+2a^2b^2-b^4+2a^2c^2+2b^2c^2-c^4]]/Sqrt[2]]; If[c

 根式里的刚好是海伦公式的表达$2 a^2 b^2+2 a^2 c^2-a^4+2 b^2 c^2-b^4-c^4=(-a+b+c) (a+b-c) (a-b+c) (a+b+c) = xyz(x+y+z)$,  也即是，$a=\frac{1}{2} \sqrt{x^2+y^2+z^2+y z+x y+x z\pm2 \sqrt{3} \sqrt{x y z (x+y+z)}}$ 如果能得到$xyz(x+y+z) =3n^2$的通解就好了。

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