{ x = 2 b y = ± ( ( 2 b ) n − 2 − b 2 ) z = ± ( ( 2 b ) n − 2 + b 2 ) {\displaystyle {\begin{cases}x=2b\\y=\pm ((2b)^{n-2}-b^{2})\\z=\pm ((2b)^{n-2}+b^{2})\end{cases}}}
{ x = 3 b y = − 9 b 3 + 4 ( 3 b ) n − 3 − 27 b 6 2 z = 9 b 3 + 4 ( 3 b ) n − 3 − 27 b 6 2 {\displaystyle {\begin{cases}x=3b\\y={\frac {-9b^{3}+{\sqrt {4(3b)^{n-3}-27b^{6}}}}{2}}\\z={\frac {9b^{3}+{\sqrt {4(3b)^{n-3}-27b^{6}}}}{2}}\end{cases}}}
{ x = b y = − 3 b 3 + 12 b n − 3 − 3 b 6 6 z = 3 b 3 + 12 b n − 3 − 3 b 6 6 {\displaystyle {\begin{cases}x=b\\y={\frac {-3b^{3}+{\sqrt {12b^{n-3}-3b^{6}}}}{6}}\\z={\frac {3b^{3}+{\sqrt {12b^{n-3}-3b^{6}}}}{6}}\end{cases}}}
{ x = 3 b y = a − 9 b 3 z = a {\displaystyle {\begin{cases}x=3b\\y=a-9b^{3}\\z=a\end{cases}}}
不等方程 ( z − y ) 3 + 3 y ( z − y ) 2 + 3 y 2 ( z − y ) = x 3 {\displaystyle (z-y)^{3}+3y(z-y)^{2}+3y^{2}(z-y)=x^{3}}
不等方程 ( z − x ) 3 + 3 x ( z − x ) 2 + 3 x 2 ( z − x ) = y 3 {\displaystyle (z-x)^{3}+3x(z-x)^{2}+3x^{2}(z-x)=y^{3}}
{ x = a b y = a − 10 a b 5 z = a {\displaystyle {\begin{cases}x=ab\\y=a-10ab^{5}\\z=a\end{cases}}}
http://mathworld.wolfram.com/EllipticCurve.html
<<School:李煌數學研究院
存在無窮多整數解,之解爲李煌解:
之全体解爲李煌解:
存在整數解,之解爲李煌解:
等價于:
存在整數解,之解爲李煌解:
之整數一定能表示為兩平方數之和，且該形式是整數能表為兩平方數之和之充要條件.
之偶數一定能表示為兩平方數之和，且該形式是偶數能表為兩平方數之和之充要條件.