We study a generalization of Pell's equation, whose coefficients are certain
algebraic integers. Let X0=0 and Xn=2+Xn−1 for each n∈Z≥1. We study the Z[Xn−1]-solutions of the
equation x2−Xn2y2=1. By imitating the solution to the classical Pell's
equation, we introduce new continued fraction expansions for Xn over
Z[Xn−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 Z2-extension over the rationals and show the
convergence of the class numbers in Z2.Comment: 17 page