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Mathematics of Computation

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



On a sequence arising in series for $ \pi $

Authors: Morris Newman and Daniel Shanks
Journal: Math. Comp. 42 (1984), 199-217
MSC: Primary 11Y35; Secondary 11F11
MathSciNet review: 725996
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Abstract: In a recent investigation of dihedral quartic fields [6] a rational sequence $ \{ {a_n}\} $ was encountered. We show that these $ {a_n}$ are positive integers and that they satisfy surprising congruences modulo a prime p. They generate unknown p-adic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding p-adic numbers are known completely [2]. Other unsolved problems are presented. The growth of the $ {a_n}$ is examined and a new algorithm for computing $ {a_n}$ is given. An appendix by D. Zagier, which carries the investigation further, is added.

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Article copyright: © Copyright 1984 American Mathematical Society