Proof, Disproof and Advances Concerning Certain Conjectures on Real Quadratic Fields
Canadian journal of mathematics, Tome 47 (1995) no. 5, pp. 1023-1036

Voir la notice de l'article provenant de la source Cambridge University Press

The purpose of this paper is to address conjectures raised in [2]. We show that one of the conjectures is false and we advance the proof of another by proving it for an infinite set of cases. Furthermore, we give hard evidence as to why the conjecture is true and show what remains to be done to complete the proof. Finally, we prove a conjecture given by S. Louboutin, for Mathematical Reviews, in his discussion of the aforementioned paper.
DOI : 10.4153/CJM-1995-054-7
Mots-clés : 11R11, 11R29, 11J70
Mollin, R. A.; Williams, H. C. Proof, Disproof and Advances Concerning Certain Conjectures on Real Quadratic Fields. Canadian journal of mathematics, Tome 47 (1995) no. 5, pp. 1023-1036. doi: 10.4153/CJM-1995-054-7
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[1] 1. Hecke, E., Eine neue art von Zetafunctionen und ihre Beziehungen zur Verteilung der Primzahlen,, Math. Z. 6(1920), 11–51. Google Scholar

[2] 2. Leu, M.G., On a Criterion for the Quadratic Fields to be of Class Number Two, Bull. London Math. Soc. 24(1992), 309–312. Google Scholar

[3] 3. Leu, M.G., On a Problem of Ono and quadratic Non-Residues, Nagoya Math. J. 115(1989), 185–198. Google Scholar

[4] 4. Mollin, R.A., Class Number One Criteria for Real Quadratic Fields I, Proc. Japan Acad. Ser. A 63(1987), 121–125. Google Scholar

[5] 5. Mollin, R.A. and Williams, H.C., Prime-Producing quadratic Polynomials and Real Quadratic Fields of Class Number One. In: Number Theory (ed. Dekoninck, J.M. and Levesque, C.), Walter de Gruyter, Berlin, 1989. 654–663. Google Scholar

[6] 6. Mollin, R.A., Solution of the Class Number One Problem for Real Quadratic Fields of Extended Richaud- Degert Type (with one possible exception). In: Number Theory (ed. Mollin, R.A.), Walter de Gruyter, Berlin, New York, 1990. 417–425. Google Scholar

[7] 7. Mollin, R.A., On a Solution of a Class Number Two Problem for a Family of Real Quadratic Fields. In: Computational Number Theory, (ed. Pethö, A. et al), Walter de Gruyter, Berlin, New York, 1991. 95–101. Google Scholar

[8] 8. Mollin, R.A., A Conjecture of S. Chowla Via the Generalized Riemann Hypothesis, Proc. Amer. Math. Soc. 102(1988), 794-796. Google Scholar

[9] 9. Mollin, R.A., Computation of the Class Number of a Real Quadratic Field, Utilitas Math. 41(1992), 259–308. Google Scholar

[10] 10. Mollin, R.A., On Real Quadratic Fields of Class Number Two, Math. Comp. 59(1992), 625–632. Google Scholar

[11] 11. Mollin, R.A., On a Determination of Real Quadratic Fields of Class Number One and Related Continued Fraction Period Length Less Than 25, Proc. Japan Acad. Ser. A. 67(1991), 20–25. Google Scholar

[12] 12. Mollin, R.A., Classification and Enumeration of Real Quadratic Fields Having Exactly One Non-Inert Prime Less Than a Minkowski Bound, Canad. Math. Bull. 36(1993), 108–115. Google Scholar

[13] 13. Mollin, R.A. and van, A.J. der Poorten, A note on symmetry and Ambiguity, Bull. Austral. Math. Soc, to appear. Google Scholar

[14] 14. Mollin, R.A., Zhang, L.-C. and Kemp, P., A Lower Bound For the Class Number of a Real Quadratic Field of ERD-type, Canad. Math. Bull. 37(1994), 90–96. Google Scholar

[15] 15. Tatuzawa, T., On a Theorem ofSiegel, Japan J. Math. 21(1951), 163–178. Google Scholar

[16] 16. Williams, H.C. and Wunderlich, M.C., On the Parallel Generation of the Residue for the Continued Fraction Factoring Algorithm, Math. Comp. 177(1987), 405–423. Google Scholar

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