回復 1# Exponential 的帖子
\(\begin{align}
& a=\sqrt[3]{2+\sqrt{\alpha }},b=\sqrt[3]{2-\sqrt{\alpha }} \\
& {{a}^{3}}+{{b}^{3}}=4,ab=\sqrt[3]{4-\alpha } \\
& {{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right) \\
& 4={{\left( a+b \right)}^{3}}-3\sqrt[3]{4-\alpha }\left( a+b \right) \\
& \alpha =4-{{\left[ \frac{{{\left( a+b \right)}^{3}}-4}{3\left( a+b \right)} \right]}^{3}}>0 \\
& {{\left[ \frac{{{\left( a+b \right)}^{3}}-4}{3\left( a+b \right)} \right]}^{3}}={{\left[ \frac{{{\left( a+b \right)}^{2}}}{3}-\frac{4}{3\left( a+b \right)} \right]}^{3}}<4 \\
& a+b\ \in \ N \\
& a+b=1\ or\ 2 \\
& \alpha =5\ or\ \frac{100}{27} \\
\end{align}\)
