四次Pell方程的一类参数特解

青青子衿

\begin{gather*}
\left\{\begin{split}
p&=18x^{6}-9(4+\sqrt{3})x^{4}+9(2+\sqrt{3})x^{2}-3-2\sqrt{3}\\
q&=3x(6x^{4}-(9+2\sqrt{3})x^{2}+2+\sqrt{3})\\
r&=3x^{2}(6x^{2}-\sqrt{3}-6)\\
s&=9x(2x^{2}-1)\\
y&=((x^{2}-1)(x^{2}-\frac{3+2\sqrt{3}}{3}))^{1/4}
\end{split}\right.\\
\\
9(97+56\sqrt{3})=\prod_{k=1}^{4}(p+i^{k}qy+i^{2k}ry^2+i^{3k}sy^3)
\end{gather*}

1. p=18x^{6}-9(4+\sqrt{3})x^{4}+9(2+\sqrt{3})x^{2}-3-2\sqrt{3}
2. q=3x(6x^{4}-(9+2\sqrt{3})x^{2}+2+\sqrt{3})
3. r=3x^{2}(6x^{2}-\sqrt{3}-6)
4. s=9x(2x^{2}-1)
5. \chi=((x^{2}-1)(x^{2}-\frac{3+2\sqrt{3}}{3}))^{1/4}
6. h=\chi^{4}
7. \frac{\left(p^{2}+hr^{2}-2hqs\right)^{2}-h\left(2pr-hs^{2}-q^{2}\right)^{2}}{9\left(97+56\sqrt{3}\right)}

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青青子衿

\begin{gather*}
\left\{\begin{split}
p&=3x^{4}-2(3+\sqrt{3})x^{2}+5+3\sqrt{3}\\
q&=x(3x^{2}+\sqrt{3})\\
r&=3x^{2}+3+2\sqrt{3}\\
s&=3x\\
y&=((x^{2}-1)(x^{2}-\tfrac{3+2\sqrt{3}}{3}))^{1/4}
\end{split}\right.\\
\\
97+56\sqrt{3}=\prod_{k=1}^{4}(p+i^{k}qy+i^{2k}ry^2+i^{3k}sy^3)
\end{gather*}

1. p=3x^{4}-2(3+\sqrt{3})x^{2}+5+3\sqrt{3}
2. q=x(3x^{2}+\sqrt{3})
3. r=3x^{2}+3+2\sqrt{3}
4. s=3x
5. \chi=((x^{2}-1)(x^{2}-\frac{3+2\sqrt{3}}{3}))^{1/4}
6. h=\chi^{4}
7. \frac{\left(p^{2}+hr^{2}-2hqs\right)^{2}-h\left(2pr-hs^{2}-q^{2}\right)^{2}}{97+56\sqrt{3}}

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