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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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Some periodic continued fractions with long periods
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by C. D. Patterson and H. C. Williams PDF
Math. Comp. 44 (1985), 523-532 Request permission

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
  • © Copyright 1985 American Mathematical Society
  • Journal: Math. Comp. 44 (1985), 523-532
  • MSC: Primary 11Y65; Secondary 11A55, 11J70
  • DOI: https://doi.org/10.1090/S0025-5718-1985-0777283-1
  • MathSciNet review: 777283