Louboutin, S.; Mollin, R. A.; Williams, H. C. Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue covers. (English) Zbl 0771.11039 Can. J. Math. 44, No. 4, 824-842 (1992). The authors prove various results concerning the connection between prime-producing quadratic polynomials and quadratic number fields with class number one. Let \(d\) be a positive square-free integer, \(f_ d(X)=- X^ 2+X+(d-1)/4\) or \(f_ d(X)=-X^ 2+d\) and \(\Delta=d\) or \(\Delta=4d\), according as \(d\equiv 1\bmod 4\) or \(d\not\equiv 1\bmod 4\). Then the main result of the paper asserts the equivalence of the following three conditions: (1) No prime \(p<\sqrt{\Delta}/2\) splits \(\mathbb{Q}(\sqrt{d})\); (2) If \(p<\sqrt{\Delta}/2\) is a prime and \(1\leq x<\sqrt{\Delta}/2\) satisfies \(f_ d(x)\equiv 0\bmod p\), then \(p\mid\Delta\); (3) \(\Delta\) is a discriminant of extended Richaud-Degert-type. Reviewer: Franz Halter-Koch (Graz) Cited in 1 ReviewCited in 11 Documents MSC: 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values 11A41 Primes Keywords:prime-producing quadratic polynomials; quadratic number fields; class number one PDFBibTeX XMLCite \textit{S. Louboutin} et al., Can. J. Math. 44, No. 4, 824--842 (1992; Zbl 0771.11039) Full Text: DOI