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Quadratic Non-Residues and Prime-Producing Polynomials

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin  and
H. C. Williams
R. A. Mollin
Affiliation:
Mathematics Dept., University of Calgary, Calgary, Alberta, T2N 1N4, Canada
H. C. Williams
Affiliation:
Computer Science Dept., University of Manitoba, Winnipeg, Manitoba, R3T2N2, Canada
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Abstract

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We will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

Keywords

12A2012A25

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 4 , 01 December 1989 , pp. 474 - 478
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Euler, E., Mem de Berlin, année 1722, 36; Comm. Arith. 1, 384.Google Scholar
2. Goldfeld, D., Gauss’ Glass Number Problem for Imaginary Quadratic Fields, Bull. Amer. Math. Soc. (New Series) 13 (1985), 23–37.Google Scholar
3. Mollin, R., Necessary and Sufficient Conditions for the Class Number of a Real Quadratic Field to be One, and a Conjecture of S. Chowla, Proceedings Amer. Math. Soc. 102 (1988), 17–21.Google Scholar
4. Mollin, R., Class Number One Criteria for Real Quadratic Fields I, Proceedings Japan Acad. Ser. A. 63 (1987), 121–125.Google Scholar
5. Mollin, R. and H. Williams, A Conjecture of S. Chowla via the Generalized Riemann Hypothesis, Proceedings Amer. Math. Soc. 102 (1988), 794–796.Google Scholar
6. Mollin, R. and H. Williams, On Prime-Valued Polynomials and Class Numbers of Real Quadratic Fields, Nagoya Math. J. 112 (1988), 143–151.Google Scholar
7. Mollin, R. and H. Williams, Prime Producing Quadratic Polynomials and Real Quadratic Fields of Class Number One (to appear: Proceedings of the International Number Theory Conference at Quebec City, July 1987).Google Scholar
8. Oesterlé, Versions effectives du theorem de Chebotarev sous LHypothèse de Riemann Généralisé, Soc. Math. France Astérisque 61 (1979), 165–167.Google Scholar
9. Patterson, C. D. and H. C. Williams, A report on the University of Manitoba Sieve Unit, Congresses Numerantium 37 (1983), 85–98.Google Scholar
10. Rosser, J. B. and L. Schoenfeld, Approximate Formulas for Some Functions of Prime Numbers, Illinois J. Math. 6 (1962), 64–94.Google Scholar