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发表于 2025-6-20 13:23:34
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显示全部楼层
绕来绕去的,差点晕了, 我发现其实就是这么一个通解,设三边是$(a,b,c)=(x^2,y^2,z^2)$,那么存在$[x^2 : y^2 : z^2] = [s^2 t^2+1 : (t+1)(1-s^2 t) : (s^2+1) t]$, 此时面积就是 $A= d^2 s t (1 + t) (1-s^2 t)$, 如果$x^2 = d(s^2 t^2+1)$
我们要解的就是这么一个通解, d是公约数,是调节因子.s,t是任意有理数, 使得
\[x^2 = d(s^2 t^2+1) \\
y^2 = d (t+1)(1-s^2 t) \\
z^2 = d (s^2+1) t \]
两个特例都经过了验证
- Table[{p,Select[Solve[{d(1+s^2 t^2)==x^2 ,d(1+t)(1-s^2 t)==y^2 ,d t(s^2+1)==z^2 ,A== d^2 s t (1+t) (1-s^2 t)}/.Thread[{x,y,z}->p],{d,s,t,A}],Last[#[[-1]]]>0&]},{p,Permutations[{11789,68104,68595},{3}]}]//Column
复制代码
\[\begin{array}{l}
\left\{\{1853,4380,4427\},\left\{\left\{d\to \frac{8306217}{5},s\to \frac{3}{34},t\to \frac{24548}{2097},A\to 32918611718880\right\}\right\}\right\} \\
\left\{\{1853,4427,4380\},\left\{\left\{d\to \frac{482770097}{233},s\to \frac{3}{34},t\to \frac{48960}{5329},A\to 32918611718880\right\}\right\}\right\} \\
\left\{\{4380,1853,4427\},\left\{\left\{d\to \frac{1874703404880}{201977},s\to \frac{4320}{5329},t\to \frac{1923769}{1509840},A\to 32918611718880\right\}\right\}\right\} \\
\left\{\{4380,4427,1853\},\left\{\left\{d\to \frac{4435433280}{233},s\to \frac{4320}{5329},t\to \frac{5329}{48960},A\to 32918611718880\right\}\right\}\right\} \\
\left\{\{4427,1853,4380\},\left\{\left\{d\to \frac{2388661245233}{201977},s\to \frac{722}{699},t\to \frac{1509840}{1923769},A\to 32918611718880\right\}\right\}\right\} \\
\left\{\{4427,4380,1853\},\left\{\left\{d\to \frac{97234628}{5},s\to \frac{722}{699},t\to \frac{2097}{24548},A\to 32918611718880\right\}\right\}\right\} \\
\end{array}
\]
.
\[
\begin{array}{l}
\left\{\{11789,68104,68595\},\left\{\left\{d\to \frac{625347242521}{17026},s\to \frac{257}{19729},t\to \frac{143819505}{1122833},A\to 284239560530875680\right\}\right\}\right\} \\
\left\{\{11789,68595,68104\},\left\{\left\{d\to \frac{2374227967609}{22865},s\to \frac{257}{19729},t\to \frac{380192}{8513},A\to 284239560530875680\right\}\right\}\right\} \\
\left\{\{68104,11789,68595\},\left\{\left\{d\to \frac{170354594418600736}{138980521},s\to \frac{896416}{1540853},t\to \frac{103049865}{35930656},A\to 284239560530875680\right\}\right\}\right\} \\
\left\{\{68104,68595,11789\},\left\{\left\{d\to \frac{106033417063456}{22865},s\to \frac{896416}{1540853},t\to \frac{8513}{380192},A\to 284239560530875680\right\}\right\}\right\} \\
\left\{\{68595,11789,68104\},\left\{\left\{d\to \frac{488580502314418065}{138980521},s\to \frac{1319445}{790789},t\to \frac{35930656}{103049865},A\to 284239560530875680\right\}\right\}\right\} \\
\left\{\{68595,68104,11789\},\left\{\left\{d\to \frac{80098403656185}{17026},s\to \frac{1319445}{790789},t\to \frac{1122833}{143819505},A\to 284239560530875680\right\}\right\}\right\} \\
\end{array}
\] |
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