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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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