On prime valued polynomials and class numbers of real quadratic fields. (English) Zbl 0629.12004

For an arbitrary positive square-free integer d we provide three sufficient conditions for the class number h(d) of \({\mathbb{Q}}(\sqrt{d})\) to equal 1. Under a certain hypothesis these conditions are shown to be necessary and sufficient. One of the conditions is that if \(f_ d(x)=- x^ 2+x+(d-1)/4\) when \(d\equiv 1\) (mod 4), and \(f_ d(x)=d-x^ 2\) when \(d\not\equiv 1\) (mod 4), then \(f_ d(x)\) is prime for all integers x such that \(1<x<\alpha\), where \(\alpha =\sqrt{d-1}/2\) when \(d\equiv 1\) (mod 4), and \(\alpha =\sqrt{d}\) if \(d\not\equiv 1\) (mod 4).
Under the assumption of the generalized Riemann hypothesis (G.R.H.) we establish that each of the three conditions is equivalent to \(h(d)=1\) if and only if d is one of the 19 values below. The latter result establishes (modulo G.R.H.) conjectures of S. Chowla, R. Mollin, and H. Yokoi. A consequence of our main result (modulo G.R.H.) is that \(f_ d(x)\) is prime for all integers x with \(1<x<\alpha\) if and only if \[ d\quad \in \quad \{2,\quad 3,\quad 5,\quad 6,\quad 7,\quad 11,\quad 13,\quad 17,^ 21,^ 29,\quad 37,\quad 53,\quad 77,\quad 101,\quad 173,\quad 197,^ 293,\quad 437,\quad 677\}\quad. \] This may be viewed as a general analog of the well-known Rabinovitch result for complex quadratic fields.


11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R09 Polynomials (irreducibility, etc.)
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
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