Solution of a problem of Yokoi
For any rational prime \(p\equiv 1 (mod 4)\), let \(\epsilon_ p=(t_ p+u_ p\sqrt{p})/2>1\) and \(h_ p\) be the fundamental unit and the class number of \({\mathbb{Q}}(\sqrt{p})\) respectively. Then, the reviewer [Investigations in number theory, Adv. Stud. Pure Math. 15, 493-501 (1988; Zbl 0664.10002)] defined new p-invariants \(n_ p\) by \(| t_ p/u^ 2_ p-n_ p| <1/2,\) and proved that there exist exactly 30 primes \(p\equiv 1 (mod 4)\) satisfying \(n_ p\neq 0\) (i.e. \(t_ p/u^ 2_ p>1/2)\) and \(h_ p=1\) with one more possible value of p. [The fundamental unit and class number one problem of real quadratic fields with prime discriminant, Nagoya Math. J. 129, 51-59 (1990).] Furthermore, \textit{Y. Taniyama} [same Journal above mentioned] added 24 non-prime square-free values of \(d\equiv 1 (mod 4)\) satisfying \(n_ d\neq 0\) and \(h_ d=1.\) In this paper, the authors add moreover 14 values of \(d\equiv 2,3 (mod 4)\) satisfying \(n_ d\neq 0\) and \(h_ d=1\). Among them, 8 values (3,7,11,23,47,83,167,227) are primes \(\equiv 3 (mod 4)\), and the remainders (2,6,14,38,62,398) are twice primes \(\equiv 3 (mod 4)\) except for 2.
scientific article; zbMATH DE number 4156570
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