论文标题
广义的佩尔方程和韦伯的班级问题问题
Generalized Pell's equations and Weber's class number problem
论文作者
论文摘要
我们研究了佩尔方程的概括，其系数是某些代数整数。令$ x_0 = 0 $和$ x_n = \ sqrt {2+x_ {n-1}} $ in \ mathbb {z} _ {\ ge 1} $。我们研究$ \ mathbb {z} [x_ {n-1}] $ - 方程的解决方案$ x^2-x_n^2y^2y^2 = 1 $。通过模仿经典佩尔方程的解决方案，我们引入了$ x_n $ over $ \ mathbb {z}的新的持续分数扩展，[x_ {n-1}] $，并获得广义pell方程的明确解决方案。此外，我们表明我们的显式解决方案在且仅当韦伯班级编号问题的答案是肯定的时，就会生成所有解决方案。我们还获得了$ \ mathbb {z} _2 $ - extension的班级编号比率的一致关系，并在$ \ mathbb {z} _2 $中显示了类数字的融合。
We study a generalization of Pell's equation, whose coefficients are certain algebraic integers. Let $X_0=0$ and $X_n=\sqrt{2+X_{n-1}}$ for each $n\in \mathbb{Z}_{\ge 1}$. We study the $\mathbb{Z}[X_{n-1}]$-solutions of the equation $x^2-X_n^2y^2=1$. By imitating the solution to the classical Pell's equation, we introduce new continued fraction expansions for $X_n$ over $\mathbb{Z}[X_{n-1}]$ and obtain an explicit solution of the generalized Pell's equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber's class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the $\mathbb{Z}_2$-extension over the rationals and show the convergence of the class numbers in $\mathbb{Z}_2$.
