On a sequence arising in series for

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 was encountered. We show that these 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 is examined and a new algorithm for computing is given. An appendix by D. Zagier, which carries the investigation further, is added.

**[1]**William Adams and Daniel Shanks,*Strong primality tests that are not sufficient*, Math. Comp.**39**(1982), no. 159, 255–300. MR**658231**, 10.1090/S0025-5718-1982-0658231-9**[2]**William Adams & Daniel Shanks, "Strong primality tests. II--Algebraic identification of the*p*-adic limits and their implications." (To appear.)**[3]**Heinrich Behnke and Friedrich Sommer,*Theorie der analytischen Funktionen einer komplexen Veränderlichen.*, Zweite veränderte Auflage. Die Grundlehren der mathematischen Wissenschaften, Bd. 77, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). MR**0147622****[4]**Marvin I. Knopp,*Modular functions in analytic number theory*, Markham Publishing Co., Chicago, Ill., 1970. MR**0265287****[5]**D. H. Lehmer and Emma Lehmer,*Cyclotomy with short periods*, Math. Comp.**41**(1983), no. 164, 743–758. MR**717718**, 10.1090/S0025-5718-1983-0717718-1**[6]**Daniel Shanks,*Dihedral quartic approximations and series for 𝜋*, J. Number Theory**14**(1982), no. 3, 397–423. MR**660385**, 10.1016/0022-314X(82)90075-0**[7]**Daniel Shanks, "Review of A. O. L. Atkin's table,"*Math. Comp.*, v. 32, 1978, p. 315.**[8]**Thomas R. Parkin and Daniel Shanks,*On the distribution of parity in the partition function*, Math. Comp.**21**(1967), 466–480. MR**0227126**, 10.1090/S0025-5718-1967-0227126-9**[9]**Daniel Shanks and Larry P. Schmid,*Variations on a theorem of Landau. I*, Math. Comp.**20**(1966), 551–569. MR**0210678**, 10.1090/S0025-5718-1966-0210678-1**[10]**Daniel Shanks, "Review 112",*Math. Comp.*, v. 19, 1965, pp. 684-686.

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

Article copyright:
© Copyright 1984
American Mathematical Society