On a sequence arising in series for
Authors:
Morris Newman and Daniel Shanks
Journal:
Math. Comp. 42 (1984), 199217
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 padic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding padic 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
(84c:10007), http://dx.doi.org/10.1090/S00255718198206582319
 [2]
William Adams & Daniel Shanks, "Strong primality tests. IIAlgebraic identification of the padic 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, SpringerVerlag,
BerlinGöttingenHeidelberg, 1962 (German). MR 0147622
(26 #5137)
 [4]
Marvin
I. Knopp, Modular functions in analytic number theory, Markham
Publishing Co., Chicago, Ill., 1970. MR 0265287
(42 #198)
 [5]
D.
H. Lehmer and Emma
Lehmer, Cyclotomy with short periods,
Math. Comp. 41 (1983), no. 164, 743–758. MR 717718
(84j:10048), http://dx.doi.org/10.1090/S00255718198307177181
 [6]
Daniel
Shanks, Dihedral quartic approximations and series for
𝜋, J. Number Theory 14 (1982), no. 3,
397–423. MR
660385 (83k:12010), http://dx.doi.org/10.1016/0022314X(82)900750
 [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
(37 #2711), http://dx.doi.org/10.1090/S00255718196702271269
 [9]
Daniel
Shanks and Larry
P. Schmid, Variations on a theorem of Landau.
I, Math. Comp. 20 (1966), 551–569. MR 0210678
(35 #1564), http://dx.doi.org/10.1090/S00255718196602106781
 [10]
Daniel Shanks, "Review 112", Math. Comp., v. 19, 1965, pp. 684686.
 [1]
 William Adams & Daniel Shanks, "Strong primality tests that are not sufficient," Math. Comp., v. 39, 1982, pp. 255300. MR 658231 (84c:10007)
 [2]
 William Adams & Daniel Shanks, "Strong primality tests. IIAlgebraic identification of the padic 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. 743758. MR 717718 (84j:10048)
 [6]
 Daniel Shanks, "Dihedral quartic approximations and series for ," J. Number Theory, v. 14, 1982, pp. 397423. 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. 446480. MR 0227126 (37:2711)
 [9]
 Daniel Shanks & Larry P. Schmid, "Variations on a theorem of Landau," Math. Comp., v. 20, 1966, pp. 551569. MR 0210678 (35:1564)
 [10]
 Daniel Shanks, "Review 112", Math. Comp., v. 19, 1965, pp. 684686.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407259969
PII:
S 00255718(1984)07259969
Article copyright:
© Copyright 1984
American Mathematical Society
