A counterexample to a conjecture of Mahler on best $p$-adic Diophantine approximation constants
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- by Alice A. Deanin PDF
- Math. Comp. 45 (1985), 621-632 Request permission
Abstract:
In 1940, Mahler proposed a conjecture regarding the value of best P-adic Diophantine approximation constants. In this paper, a computational technique which tests the conjecture for any particular P is described. A computer search verified the conjecture for all $P \leqslant 101$, except 83. The case P = 83 is discussed. A counterexample is given.References
- H. Davenport, Note on linear fractional substitutions with large determinant, Ann. of Math. (2) 41 (1940), 59β62. MR 1771, DOI 10.2307/1968820 A. A. Deanin, Mahlerβs P-Adic Continued Fraction Algorithm, Ph.D. Dissertation, University of Maryland, 1983.
- Alice A. Deanin, Periodicity of $P$-adic continued fraction expansions, J. Number Theory 23 (1986), no.Β 3, 367β387. MR 846967, DOI 10.1016/0022-314X(86)90082-X L. R. Ford, Automorphic Functions, 2nd ed., Chelsea, New York, 1951.
- Kurt Mahler, On a geometrical representation of $p$-adic numbers, Ann. of Math. (2) 41 (1940), 8β56. MR 1772, DOI 10.2307/1968818 B. M. M. de Weger, Approximation Lattices of p-Adic Numbers. Report no. 22, Math. Inst. R. U. Leiden, 1984.
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp. 45 (1985), 621-632
- MSC: Primary 11J61
- DOI: https://doi.org/10.1090/S0025-5718-1985-0804950-3
- MathSciNet review: 804950