On the fundamental units and the class numbers of real quadratic fields. II (Q1112108)

[For part I see Nagoya Math. J. 95, 125-135 (1984; Zbl 0533.12008).] Suppose m is a positive square free integer, and \({\mathbb{Q}}(\sqrt{m})\) is a real quadratic field with discriminant D, class number h(m) and fundamental unit \(\epsilon_ m\). The author studies the problem of obtaining lower bounds for h(m) and \(\epsilon_ m\). For example, if \[ m=(C^ s+\mu (C^ t- \lambda))^ 2+4\lambda C^ t,\quad s>t\geq 1;\quad \lambda,\mu =\pm 1, \] and C is even and not a power of 2, then \(h(m)>(\log (D)/\log (4))-1\), improving work of Lang, Takeuchi and the author. The author also gives \(\epsilon_ m\) explicitly for such m.