Class Numbers of Real Quadratic Fields, Continued Fractions, Reduced Ideals, Prime-Producing Quadratic Polynomials and Quadratic Residue Covers

Louboutin, S. ; Mollin, R. A. ; Williams, H. C.

Canadian journal of mathematics, Tome 44 (1992) no. 4, pp. 824-842

Résumé

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

Détail

DOI : 10.4153/CJM-1992-049-0

Mots-clés : 11R11, 11R09, 11R29

Bibliographie

[1] 1. Cohn, H., A second course in number theory, John Wiley and Sons Inc., New York/London (1962). Google Scholar

[2] 2. Dickson, L.E., Theory of Numbers, Chelsea, NY (1957). Google Scholar

[3] 3. Friesen, C., Legendre symbols and continued fractions, Acta. Arith. LIX(1991), 365–379. Google Scholar

[4] 4. Halter-Koch, F., Prime-producing quadratic polynomials and class numbers of quadratic orders. In: Computational Number Theory, (Pethô, A.,Pohst, M., H.Williams, C. and H.Zimmer, G., eds.), Walter de Gruyter, Berlin (1991), 73–82. Google Scholar

[5] 5. Hardy, G.H. and Littlewood, J.E., Some problems of partitio numerorium; HI: On the expression of a number as a sum of primes, Acta. Math. (1924), 1–70. Google Scholar

[6] 6. Hensley, D. and Richards, I., On the incompatibility of two conjectures concerning primes, Proc. Symp. in Pure Math., Analytic Number Theory (AMS) 24 (1973), 123–127. Google Scholar

[7] 7. Hua, L.K., Introduction to number theory, Springer-Verlag (1982). Google Scholar

[8] 8. Kaplan, P. and Williams, K.S., The distance between ideals in orders of real quadratic fields, L'Enseignement Math. 36 (1990), 321–358. Google Scholar

[9] 9. Lauchaud, G., Sur les corps quadratiques reels principaux, Séminaire de théorie de nombres; Paris 1984–85. Progress in Math. 63,165–175. Google Scholar

[10] 10. Louboutin, S., Groupes des classes d'idéaux triviaux, Acta. Arith, LIV(1989), 61–74. Google Scholar

[11] 11. Louboutin, S., Continued Fractions and Real Quadratic Fields, J. Number Theory, 30 (1988), 167–176. Google Scholar

[12] 12. Louboutin, S., Prime producing quadratic polynomials and class numbers of real quadratic fields, Canad. J. Math. XLII(1990), 315–341. Google Scholar

[13] 13. Louboutin, S., Extensions du Théorème de Frobenius-Rabinowitsch,C.R. Acad. Sci. Paris 1312 (1991),711–714. Google Scholar

[14] 14. Lu, H., On the Class-Number of Real Quadratic Fields, Sci. Sinica, (Special issue) 2 (1979), 118–130. Google Scholar

[15] 15. Mollin, R.A., Necessary and Sufficient Conditions for the Class Number of a Real Quadratic Field to be One and a Conjecture of Chowla S., Proc. Amer. Math. Soc. 102 (1988), 17–21. Google Scholar

[16] 16. Mollin, R.A., Lower Bounds for Class Numbers of Real Quadratic and Biquadratic Fields, Proceed. Amer. Math. Soc. 101 (1987), 439–444. Google Scholar

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

[18] 18. Mollin, R.A., Class Number One Criteria for Real Quadratic Fields II, Proc. Japan Acad. (A) 63 (1987), 162- 164. Google Scholar

[19] 19. Mollin, R.A., On the Insolubility of a class diophantine equations and the non-triviality of the class number of related real quadratic fields ofRichaud-Degerttype, NagoyaMath. J. 105 (1987), 39–47. Google Scholar

[20] 20. Mollin, R.A. and Williams, H.C., Computation of real quadratic fields with class number one, Advances in the theory of computation and computational math., to appear. Google Scholar

[21] 21. Mollin, R.A., Solution of the class number one problem for real quadratic fields of Richaud-Degert type (with one possible exception). In: Number Theory, Walter de Gruyter and Co., Berlin, (1990), (Mollin, R.A., éd.), 417–425. Google Scholar

[22] 22. Mollin, R.A., Consecutive powers in continued fractions, Acta. Arith., to appear. Google Scholar

[23] 23. Mollin, R.A., Prime-producing polynomials and real quadratic fields of class number one. In: Number Theory, (Levesque, C. and DeKoninck, J.M. (eds.)), Walter de Gruyter and Co., (1989), 654–663. Google Scholar

[24] 24. Mollin, R.A., Class Number one for real quadratic fields, continued fractions, and reduced ideals. In: Number Theory and Applications, (Mollin, R.A., ed.) NATO ASI, C265 (1989), 481–496. Google Scholar

[25] 25. Mollin, R.A., Period four and real quadratic fields of class number one, Proc. Japan Acad. (A) 65 (1989), 89–93. Google Scholar

[26] 26. Mollin, R.A., Class number problems for real quadratic fields. In: Number Theory and Cryptography, London Math. Soc. Lecture Note Series, 154 (1990), 177–195. Google Scholar

[27] 27. Mollin, R.A., Real quadratic fields of class number one and continued faction period less than six, C.R. Math. Rep. Acad. Sci. Canada XI( 1989), 51–56. Google Scholar

[28] 28. Mollin, R.A., Powers of two, continued fractions, and real quadratic fields of class number one, Memorial volume to Gauss, C.F. (Rassias, G. (ed.)), to appear. Google Scholar

[29] 29. Mollin, R.A., On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J. 112 (1988), 143–151. Google Scholar

[30] 30. Mollin, R.A., Affirmative solution of a conjecture related to a sequence of Shanks, Proc. Japan Acad. (A) 67 (1991), 70–72. Google Scholar

[31] 31. Mollin, R.A., Continued fractions of period five and real quadratic fields of class number one, Acta. Arth. LVI(1990), 55–63. Google Scholar

[32] 32. Mollin, R.A., Quadratic Residue Covers for Real Quadratic Fields, to appear. Google Scholar

[33] 33. Rabinowitsch, G., Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratishen Zahlkorpern, Proc. Fifth Internat. Congress Math. (Cambridge) (1913), 418–421. Google Scholar

[34] 34. Mollin, R.A., Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratishen Zahlkorpern, J. Rein. Angew. Math. 142 (1913), 153–164. Google Scholar

[35] 35. Ricci, G., Ricerche arithmetiche suipolinomi, Rend. Circ. Math. Palermo 57 (1933), 433–475. Google Scholar

[36] 36. Shanks, D., The infrastructure of real quadratic fields and its applications, Proc. 1972 number theory Conf., Boulder, CO (1973), 217–224. Google Scholar

[37] 37. Tatuzawa, T., On a theorem of Siegel, Japan J. Math. 21 (1951), 163–178. Google Scholar

[38] 38. Williams, H.C., Continued fractions and number theoretic computations, Rocky Mtn. J. Math. 15 (1985), 621–655. Google Scholar

[39] 39. Williams, H.C. and Wunderlich, M.C., On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 177 (1987), 405–423. Google Scholar