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Prime producing polynomials: Proof of a conjecture by Mollin and Williams. (English) Zbl 0927.11051

The genesis of the problem concerning class number one and prime producing quadratic polynomials was the 1913 Rabinowitsch result, which may be stated as follows. Given a squarefree negative discriminant \(\Delta= 1-4A\) of a complex quadratic field \(\mathbb{Q}(\sqrt{\Delta})\), with class number \(h_\Delta\), \(x^2+x+A\) is prime for \(x=0,1,\dots, A-2\) if and only if \(h_\Delta=1\). More recent research has focused upon the real quadratic field analogue of this result. In particular, this reviewer and H. C. Williams found, what the author under review calls, the prototype result in this direction given as follows.
If \(\Delta= 4A+1\) is the squarefree discriminant of \(\mathbb{Q} (\sqrt{\Delta})\), then \(A-x^2-x\) is prime for any positive \(n< \sqrt{a}-1\) if and only if \(h_\Delta=1\) and either \(\Delta=17\) or \(\Delta\geq 21\) and \(\Delta\equiv 1\pmod 8\) with \(\Delta\) of the form \(4m^2+1\) or \(m^2\pm 4\).
Numerous related results have ensued [see this reviewer’s book (*) Quadratics, CRC Press (1996; Zbl 0858.11001) for complete details]. In particular, a conjecture is posed on page 140 of (*), which the paper under review establishes as valid.
The paper is elegant, concise and easy to read.
Reviewer: R.Mollin (Calgary)

MSC:

11R09 Polynomials (irreducibility, etc.)
11R29 Class numbers, class groups, discriminants
11E16 General binary quadratic forms
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values

Citations:

Zbl 0858.11001
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