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Prime-producing polynomials related to class number one problem of number fields. (English) Zbl 1525.11127

The author first recalls some results for prime-producing polynomials which are related to class number one problem of quadratic fields. These characterize polynomials of the form \(f_q(x)=x^2+x+q\) and \(g_q(x)=-x^2+x+q^2\) for which the numbers \(f_q(x)\), \(x=0,1,\dots,x-2\), and \(g_q(x)\), \(x=2,3,\dots,q\), are all prime. For \(f_q\) this happens if and only if \(q=2,3,5,11,17,41\) exactly when the number field \({\mathbb Q}(\sqrt{1-4q})\) has class number one. Then, he makes a conjecture concerning the primality of the values of the polynomial \(x^3+mx^2-(m+3)x+1\) at the values \(x=-m-1,-m,\dots,-1\). Finally, he compares numerically the ratios producing prime values for several polynomials in some interval.

MSC:

11R09 Polynomials (irreducibility, etc.)
11R11 Quadratic extensions
11R16 Cubic and quartic extensions
11R29 Class numbers, class groups, discriminants
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