Effective irrationality measures for certain algebraic numbers
Author:
David Easton
Journal:
Math. Comp. 46 (1986), 613622
MSC:
Primary 11J68; Secondary 11J82
MathSciNet review:
829632
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Abstract: A result of Chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given. More particularly, Padé approximants to the function are employed to show, for certain integers a and b, that Here, c and k are given as functions of a and b only.
 [1]
A.
Baker, Rational approximations to \root3\of2 and other algebraic
numbers, Quart. J. Math. Oxford Ser. (2) 15 (1964),
375–383. MR 0171750
(30 #1977)
 [2]
G.
V. Chudnovsky, On the method of ThueSiegel, Ann. of Math. (2)
117 (1983), no. 2, 325–382. MR 690849
(85g:11058), http://dx.doi.org/10.2307/2007080
 [3]
G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford Univ. Press, London, 1960.
 [4]
William
Judson LeVeque, Topics in number theory. Vols. 1 and 2,
AddisonWesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
(18,283d)
 [5]
Kevin
S. McCurley, Explicit estimates for the error term
in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004
(85e:11065), http://dx.doi.org/10.1090/S00255718198407260046
 [6]
Kevin
S. McCurley, Explicit estimates for
𝜃(𝑥;3,𝑙) and
𝜓(𝑥;3,𝑙), Math.
Comp. 42 (1984), no. 165, 287–296. MR 726005
(85g:11085), http://dx.doi.org/10.1090/S00255718198407260058
 [7]
J.
Barkley Rosser and Lowell
Schoenfeld, Sharper bounds for the Chebyshev
functions 𝜃(𝑥) and 𝜓(𝑥), Math. Comp. 29 (1975), 243–269.
Collection of articles dedicated to Derrick Henry Lehmer on the occasion of
his seventieth birthday. MR 0457373
(56 #15581a), http://dx.doi.org/10.1090/S00255718197504573737
 [8]
Carl
Ludwig Siegel, Die Gleichung
𝑎𝑥ⁿ–𝑏𝑦ⁿ=𝑐,
Math. Ann. 114 (1937), no. 1, 57–68 (German).
MR
1513124, http://dx.doi.org/10.1007/BF01594162
 [9]
Lucy
Joan Slater, Generalized hypergeometric functions, Cambridge
University Press, Cambridge, 1966. MR 0201688
(34 #1570)
 [1]
 A. Baker, "Rational approximation to and other algebraic numbers," Quart. J. Math. Oxford Ser. (2), v. 15, 1964, pp. 375383. MR 0171750 (30:1977)
 [2]
 G. V. Chudnovsky, "On the method of ThueSiegel," Ann. of Math., v. 117, 1983, pp. 325382. MR 690849 (85g:11058)
 [3]
 G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford Univ. Press, London, 1960.
 [4]
 W. J. LeVeque, Topics in Number Theory, AddisonWesley, Reading, Mass., 1956. MR 0080682 (18:283d)
 [5]
 K. S. McCurley, "Explicit estimates for the error term in the prime number theorem for arithmetic progressions," Math. Comp., v. 42, 1984, pp. 265286. MR 726004 (85e:11065)
 [6]
 K. S. McCurley, "Explicit estimates for and ," Math. Comp., v. 42, 1984, pp. 287296. MR 726005 (85g:11085)
 [7]
 J. B. Rosser & L. Schoenfeld, "Sharper bounds for the Chebyshev functions and ," Math. Comp., v. 29, 1975, pp. 243269. MR 0457373 (56:15581a)
 [8]
 C. L. Siegel, "Die Gleichung ," Math. Ann., v. 114, 1937, pp. 5768. MR 1513124
 [9]
 L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, London, 1966. MR 0201688 (34:1570)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608296324
PII:
S 00255718(1986)08296324
Article copyright:
© Copyright 1986
American Mathematical Society
