Prime-producing quadratic polynomials and class numbers of quadratic orders. (English) Zbl 0728.11049

Computational number theory, Proc. Colloq., Debrecen/Hung. 1989, 73-82 (1991).
[For the entire collection see Zbl 0722.00004.]
The author gives necessary (but not sufficient), as well as sufficient (but not necessary) conditions for certain canonical quadratic polynomials to be prime (or 1) in an initial range of values. The underlying class group of the order under consideration has class number 1 or 2. The author uses the results to generalize earlier work of Frobenius, Hendy, Louboutin, this reviewer and H. C. Williams.
In the author’s “Remark” on page 76 he says that it is not known whether there are positive discriminants satisfying the hypothesis of Theorem 3.1 and not being of Richaud-Degert-type. The answer is that there are such values (such as 341 and 917). Also the “Remark” on page 81 should refer to Theorem 5.1 rather than 3.1. Finally, in Theorem 3.1 the author must stipulate that \(q\) does not divide \(f\) if \(q>1\). Otherwise he cannot invoke the Theorem for \(q=1\) as he does in Corollary 3.1.


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


Zbl 0722.00004