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New prime-producing quadratic polynomials associated with class number one or two. (English) Zbl 1015.11054
For a square-free positive integer $$D>1$$, set $$\Delta=D$$ if $$D\equiv 1\pmod 4$$ and $$\Delta= 4D$$ otherwise. If $$\Delta= \ell^2+r$$ is a field discriminant with $$r\mid 4\ell$$, then $$\Delta$$ is said to be of extended Richaud-Degert type (ERD-type).
In this paper, the author intends to provide necessary and sufficient conditions for a real quadratic field of ERD-type to have class number one or two in terms of some prime-producing quadratic polynomials.
Namely, in the case of $$\Delta= 4(t^2\pm 2)$$ $$(t>1)$$ he proves that the class number $$h(\Delta)=1$$ if and only if $$f_t(x)= -2x^2+ 2tx\pm 1$$ is prime for any natural number $$x<t$$, and moreover he provides a necessary and sufficient condition for either $$h(\Delta)=1$$ and $$S=\phi$$ or $$h(\Delta)=2$$, where $$S$$ means the set of all odd primes $$p$$ satisfying $$p< \sqrt{D}$$ and $$(D/p)\neq -1$$.
##### MSC:
 11R29 Class numbers, class groups, discriminants 11R11 Quadratic extensions 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
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