Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some periodic continued fractions with long periods

Authors: C. D. Patterson and H. C. Williams
Journal: Math. Comp. 44 (1985), 523-532
MSC: Primary 11Y65; Secondary 11A55, 11J70
MathSciNet review: 777283
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p(D)$ be the period length of the continued fraction for $ \sqrt D $ . Under the extended Riemann Hypothesis for $ \mathcal{Q}(\sqrt D )$ one would expect that $ p(D) = O({D^{1/2}}\log \log D)$. In order to test this it is necessary to find values of D for which $ p(D)$ is large. This, in turn, requires that we be able to find solutions to large sets of simultaneous linear congruences. The University of Manitoba Sieve Unit (UMSU), a machine similar to D. H. Lehmer's DLS-127, was used to find such values of D. For example, if $ D = 46257585588439$, then $ p(D) = 25679652$ 25679652. Some results are also obtained for the Voronoi continued fraction for $ ^3\sqrt D $.

References [Enhancements On Off] (What's this?)

  • [1] P. Barrucand & H. Cohn, "Remarks on principal factors in a relative cubic field," J. Number Theory, v. 3, 1971, pp. 226-239. MR 0276197 (43:1945)
  • [2] T. Honda, "Pure cubic fields whose class numbers are multiples of three," J. Number Theory, v. 3, 1971, pp. 7-12. MR 0292795 (45:1877)
  • [3] D. H. Lehmer, "A history of the sieve process," in A History of Computing in the Twentieth Century, Academic Press, New York, 1980, pp. 445-456.
  • [4] D. H. Lehmer, E. Lehmer & D. Shanks, "Integer sequences having prescribed quadratic character," Math. Comp., v. 24, 1970, pp. 433-451. MR 0271006 (42:5889)
  • [5] C. D. Patterson & H. C. Williams, "A report on the University of Manitoba Sieve Unit," Congr. Numer., v. 37, 1983, pp. 85-98. MR 703580 (84g:10003)
  • [6] G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
  • [7] H. C. Williams, "A numerical investigation into the length of the period of the continued fraction expansion of $ \sqrt D $ ," Math. Comp., v. 36, 1981, pp. 593-601. MR 606518 (82f:10011)
  • [8] H. C. Williams, Continued Fractions and Number Theoretic Computations, Proc. Number Theory Conf., Edmonton, 1983; Rocky Mountain J. Math. (To appear.) MR 823273 (87h:11129)
  • [9] H. C. Williams, G. W. Dueck & B. K. Schmid, "A rapid method of evaluating the regulator and class number of a pure cubic field," Math. Comp., v. 41, 1983, pp. 235-286. MR 701638 (84m:12010)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11Y65, 11A55, 11J70

Retrieve articles in all journals with MSC: 11Y65, 11A55, 11J70

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society