Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation |
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Authors: | Bauer Mark Bennett Michael A. |
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Affiliation: | (1) Department of Mathematics, University of Illinois, Urbana, Illinois, 61801 |
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Abstract: | In this paper, we refine work of Beukers, applying results from the theory of Padé approximation to (1 – z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to (where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D. |
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Keywords: | Ramanujan-Nagell equation hypergeometric method Padé approximation |
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