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
Full Text: DOI


[1] Atti. Acad. Pontif. Nuovi. Lincei pp 177– (1866)
[2] DOI: 10.1017/S0017089500002718 · Zbl 0323.12006 · doi:10.1017/S0017089500002718
[3] J. reine angew. Math. 142 pp 153– (1913)
[4] Nagoya Math. J. 95 pp 125– (1984) · Zbl 0533.12008 · doi:10.1017/S0027763000021036
[5] Proc. Fifth Internat. Congress Math. 1 pp 418– (1913)
[6] DOI: 10.1090/S0002-9939-1988-0934844-9 · doi:10.1090/S0002-9939-1988-0934844-9
[7] DOI: 10.1016/0022-314X(86)90053-3 · Zbl 0591.12006 · doi:10.1016/0022-314X(86)90053-3
[8] Nagoya Math. J. 105 pp 39– (1987) · Zbl 0591.12005 · doi:10.1017/S0027763000000738
[9] Proceedings Japan Acad. 83 pp 121– (1987)
[10] DOI: 10.1090/S0002-9939-1988-0915707-1 · doi:10.1090/S0002-9939-1988-0915707-1
[11] Nagoya Math. J. 79 pp 123– (1980) · Zbl 0447.12006 · doi:10.1017/S0027763000018961
[12] Proc. Int. Conf. on class numbers and fundamental units pp 125– (1986)
[13] DOI: 10.1090/S0273-0979-1985-15352-2 · Zbl 0572.12004 · doi:10.1090/S0273-0979-1985-15352-2
[14] DOI: 10.1307/mmj/1028999653 · Zbl 0148.27802 · doi:10.1307/mmj/1028999653
[15] DOI: 10.1007/BF02941943 · Zbl 0079.05803 · doi:10.1007/BF02941943
[16] DOI: 10.1112/S0025579300003971 · Zbl 0161.05201 · doi:10.1112/S0025579300003971
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.