A numerical investigation into the length of the period of the continued fraction expansion of $\sqrt {D}$
HTML articles powered by AMS MathViewer
- by H. C. Williams PDF
- Math. Comp. 36 (1981), 593-601 Request permission
Abstract:
Let $p(D)$ be the length of the period of the continued fraction expansion of $\sqrt D$, where D is a positive square-free integer. In this paper it is suggested that $p(D) = O(\sqrt D \log \log D)$ and several tables of numerical results, which support this suggestion, are provided.References
- B. D. Beach and H. C. Williams, Some computer results on periodic continued fractions, Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971) Louisiana State Univ., Baton Rouge, La., 1971, pp. 133–146. MR 0321880
- J. H. E. Cohn, The length of the period of the simple continued fraction of $d^{1/2}$, Pacific J. Math. 71 (1977), no. 1, 21–32. MR 457335, DOI 10.2140/pjm.1977.71.21
- Paul Lévy, Sur le développement en fraction continue d’un nombre choisi au hasard, Compositio Math. 3 (1936), 286–303 (French). MR 1556945 J. E. Littlewood, "On the class-number of the corpus $P(\sqrt { - k} )$," Proc. London Math. Soc., v. 28, 1928, pp. 358-372.
- Daniel Shanks, The infrastructure of a real quadratic field and its applications, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 217–224. MR 0389842
- Daniel Shanks, Systematic examination of Littlewood’s bounds on $L(1,\,\chi )$, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 267–283. MR 0337827
- Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8
- R. G. Stanton, C. Sudler Jr., and H. C. Williams, An upper bound for the period of the simple continued fraction for $\sqrt {D}$, Pacific J. Math. 67 (1976), no. 2, 525–536. MR 429724, DOI 10.2140/pjm.1976.67.525
- H. C. Williams and J. Broere, A computational technique for evaluating $L(1,\chi )$ and the class number of a real quadratic field, Math. Comp. 30 (1976), no. 136, 887–893. MR 414522, DOI 10.1090/S0025-5718-1976-0414522-5
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Math. Comp. 36 (1981), 593-601
- MSC: Primary 10A32; Secondary 10-04
- DOI: https://doi.org/10.1090/S0025-5718-1981-0606518-7
- MathSciNet review: 606518