A counterexample to a conjecture of Mahler on best -adic Diophantine approximation constants

Author:
Alice A. Deanin

Journal:
Math. Comp. **45** (1985), 621-632

MSC:
Primary 11J61

MathSciNet review:
804950

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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 , except 83. The case *P* = 83 is discussed. A counterexample is given.

**[1]**H. Davenport,*Note on linear fractional substitutions with large determinant*, Ann. of Math. (2)**41**(1940), 59–62. MR**0001771****[2]**A. A. Deanin,*Mahler's P-Adic Continued Fraction Algorithm*, Ph.D. Dissertation, University of Maryland, 1983.**[3]**Alice A. Deanin,*Periodicity of 𝑃-adic continued fraction expansions*, J. Number Theory**23**(1986), no. 3, 367–387. MR**846967**, 10.1016/0022-314X(86)90082-X**[4]**L. R. Ford,*Automorphic Functions*, 2nd ed., Chelsea, New York, 1951.**[5]**Kurt Mahler,*On a geometrical representation of 𝑝-adic numbers*, Ann. of Math. (2)**41**(1940), 8–56. MR**0001772****[6]**B. M. M. de Weger,*Approximation Lattices of p-Adic Numbers*. Report no. 22, Math. Inst. R. U. Leiden, 1984.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1985-0804950-3

Article copyright:
© Copyright 1985
American Mathematical Society