Class number one problem for real quadratic fields. (The conjecture of Gauss) (Q1113942)

Gauss conjectured that there are infinitely many real quadratic number fields o f class number 1. The author proves the following theorem: Let \[ U=\{2^{\delta}\prod_{i}p_ i^{e_ i} | \delta =0\quad or\quad 1,\quad e_ i\geq 1,\quad p_ i\text{ primes }\equiv 1\pmod 4\}. \] Then, for any fixed \(u\in U\), there exist only finitely many real quadratic fields \({\mathbb Q}(\sqrt{p})\) of class number 1 with p a prime \(\equiv 1\) (mod 4) and \(u=u_ p\), where \(\epsilon_ p=(t_ p+u_ p\sqrt{p})/2\) is the fundamental unit of \({\mathbb Q}(\sqrt{p})\).