Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue covers. (English) Zbl 0771.11039

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.


11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11A41 Primes
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