On a sequence arising in series for $\pi$
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- by Morris Newman and Daniel Shanks PDF
- Math. Comp. 42 (1984), 199-217 Request permission
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
- William Adams and Daniel Shanks, Strong primality tests that are not sufficient, Math. Comp. 39 (1982), no. 159, 255–300. MR 658231, DOI 10.1090/S0025-5718-1982-0658231-9 William Adams & Daniel Shanks, "Strong primality tests. II—Algebraic identification of the p-adic limits and their implications." (To appear.)
- Heinrich Behnke and Friedrich Sommer, Theorie der analytischen Funktionen einer komplexen Veränderlichen. , Die Grundlehren der mathematischen Wissenschaften, Band 77, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). Zweite veränderte Auflage. MR 0147622, DOI 10.1007/978-3-662-01316-8
- Marvin I. Knopp, Modular functions in analytic number theory, Markham Publishing Co., Chicago, Ill., 1970. MR 0265287
- D. H. Lehmer and Emma Lehmer, Cyclotomy with short periods, Math. Comp. 41 (1983), no. 164, 743–758. MR 717718, DOI 10.1090/S0025-5718-1983-0717718-1
- Daniel Shanks, Dihedral quartic approximations and series for $\pi$, J. Number Theory 14 (1982), no. 3, 397–423. MR 660385, DOI 10.1016/0022-314X(82)90075-0 Daniel Shanks, "Review of A. O. L. Atkin’s table," Math. Comp., v. 32, 1978, p. 315.
- Thomas R. Parkin and Daniel Shanks, On the distribution of parity in the partition function, Math. Comp. 21 (1967), 466–480. MR 227126, DOI 10.1090/S0025-5718-1967-0227126-9
- Daniel Shanks and Larry P. Schmid, Variations on a theorem of Landau. I, Math. Comp. 20 (1966), 551–569. MR 210678, DOI 10.1090/S0025-5718-1966-0210678-1 Daniel Shanks, "Review 112", Math. Comp., v. 19, 1965, pp. 684-686.
Additional Information
- © Copyright 1984 American Mathematical Society
- 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