On bounds for fundamental units of real quadratic fields (Q1323879)

The authors prove a theorem, which has as a consequence the equivalence of the following two conjectures: (1) There exist infinitely many real quadratic fields \(\mathbb{Q} (\sqrt{D})\) with class number 1. (2) For any given \(M>0\), there exists at least one real quadratic field \(\mathbb{Q}( \sqrt{D})\) of class number one such that the fundamental unit \(\varepsilon_ D\) of \(\mathbb{Q} (\sqrt{D})\) satisfies \(\varepsilon_ D > f(M,D)\), where \(f\) is a function satisfying certain conditions.