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Mathematics of Computation
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
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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 $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1985-0777283-1
PII: S 0025-5718(1985)0777283-1
Article copyright: © Copyright 1985 American Mathematical Society