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

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

MathSciNet review:
725996

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Abstract | References | Similar Articles | Additional Information

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 & 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 ,"*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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DOI:
https://doi.org/10.1090/S0025-5718-1984-0725996-9

Article copyright:
© Copyright 1984
American Mathematical Society