李煌方程: ( 2 2 k − 1 − 2 ) 2 z k = x 4 − y 2 {\displaystyle ({2^{2k-1}-2})^{2}z^{k}=x^{4}-y^{2}} 之通解公式
{ x = t 2 k ( 1 + 2 2 k − 2 ) y = 2 2 k t 4 k − t 4 k ( 2 2 k − 2 − 1 ) 2 z = 4 t 8 {\displaystyle {\begin{cases}x=t^{2k}(1+2^{2k-2})\\y=2^{2k}t^{4k}-t^{4k}({2^{2k-2}-1})^{2}\\z=4t^{8}\end{cases}}}
李煌方程: x 2 + a y 2 = z 4 {\displaystyle x^{2}+ay^{2}=z^{4}} 之通解公式
{ x = 4 ( a − 1 ) + 4 − ( a − 1 ) 2 = 6 a − a 2 − 1 y = 4 ( a − 1 ) = 4 a − 4 z = a + 1 {\displaystyle {\begin{cases}x=4(a-1)+4-(a-1)^{2}=6a-a^{2}-1\\y=4(a-1)=4a-4\\z=a+1\end{cases}}}
http://mathworld.wolfram.com/EllipticCurve.html
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