A note on units of real quadratic fields (Q2902036)

The authors prove that if \((t_d+u_d\sqrt{d})/\sigma \) is the fundamental unit of the real quadratic field \(\mathbb Q(\sqrt{d})\) with \(\sigma=2 \) provided \(d\equiv 1 \pmod{4}\) and \(\sigma=1\) otherwise, then \(u_d<d\) for all square-free integers \(d\) provided \(d\in S(\ell;a_1,\ldots a_{\ell-1})=\{d\in\mathbb Z\mid d>0,\sqrt{d}=[a_0,\overline{a_1, \ldots a_{\ell-1}}], \) with one possible exception, where \( a_1,\dots,a_{\ell-1} \) is a palindrome. This is then applied to the Ankeny-Artin-Chowla conjecture and the Mordell conjecture.