2020/10/13 by Hyuga Yoshizaki, Yoshizaki, Hyuga
Computer Science · Mathematics · #11D57 #11J70 #11R18 #11R27 #11R29 #Advanced Mathematical Identities #FOS: Mathematics #Functional Equations Stability Results #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2010.06399
openalex publication_date 2020/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.