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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
DOI: https://doi.org/10.1090/S0025-5718-1984-0725996-9
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.


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

  • [1] William Adams & Daniel Shanks, "Strong primality tests that are not sufficient," Math. Comp., v. 39, 1982, pp. 255-300. MR 658231 (84c:10007)
  • [2] William Adams & Daniel Shanks, "Strong primality tests. II--Algebraic identification of the p-adic limits and their implications." (To appear.)
  • [3] H. Behnke & F. Sommer, Theorie der analytischen Funktionen einer complexen Veränderlichen, Springer, Berlin, 1965, viii + 603 pp. MR 0147622 (26:5137)
  • [4] Marvin I. Knopp, Modular Functions in Analytic Number Theory, Markham, Chicago, Ill., 1970, x + 150 pp. MR 0265287 (42:198)
  • [5] Derrick H. Lehmer & Emma Lehmer, "Cyclotomy with short periods," Math. Comp., v. 41, 1983, pp. 743-758. MR 717718 (84j:10048)
  • [6] Daniel Shanks, "Dihedral quartic approximations and series for $ \pi $," J. Number Theory, v. 14, 1982, pp. 397-423. MR 660385 (83k:12010)
  • [7] Daniel Shanks, "Review of A. O. L. Atkin's table," Math. Comp., v. 32, 1978, p. 315.
  • [8] Thomas R. Parkin & Daniel Shanks, "On the distribution of parity in the partition function," Math. Comp., v. 21, 1967, pp. 446-480. MR 0227126 (37:2711)
  • [9] Daniel Shanks & Larry P. Schmid, "Variations on a theorem of Landau," Math. Comp., v. 20, 1966, pp. 551-569. MR 0210678 (35:1564)
  • [10] Daniel Shanks, "Review 112", Math. Comp., v. 19, 1965, pp. 684-686.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1984-0725996-9
Article copyright: © Copyright 1984 American Mathematical Society

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