A New Continued Fraction Expansion and Weber's Class Number Problem

Hyuga Yoshizaki

Published 2020-10-13Version 1

We give a new continued fraction expansion algorithm for certain real numbers related to Weber's class number problem. By considering the analogy of the solution of Pell's equations, we get an explicit unit. We conjecture that such a unit generates the relative units, and show that this conjecture is equivalent to Weber's class number problem. We give some partial results for our conjecture. We show that our conjecture is true for $n=2$ and the proof is independent of what Weber did. We also show that some minimality of this explicit unit. Moreover we give an estimate for the ratio of class numbers $h_n$ of $n$-th layers of $\Z_2\hi$extension of $\Q$ modulo $2^n$ as $h_n/h_{n-1}\equiv 1 \pmod{2^n}$.