## 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.

### MSC:

 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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