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Mathematics of Computation

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On the Diophantine equation $|ax^{n}-by^{n}|=1$


Authors: Michael A. Bennett and Benjamin M. M. de Weger
Journal: Math. Comp. 67 (1998), 413-438
MSC (1991): Primary 11D41; Secondary 11Y50
DOI: https://doi.org/10.1090/S0025-5718-98-00900-4
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Abstract: If $ a, b $ and $ n $ are positive integers with $ b \geq a $ and $ n \geq 3 $, then the equation of the title possesses at most one solution in positive integers $ x $ and $ y $, with the possible exceptions of $ ( a, b, n ) $ satisfying $ b = a + 1 $, $ 2 \leq a \leq \min \{ 0.3 n, 83 \} $ and $ 17 \leq n \leq 347 $. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.


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Additional Information

Michael A. Bennett
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: mabennet@math.lsa.umich.edu

Benjamin M. M. de Weger
Affiliation: Mathematical Institute, University of Leiden, Leiden, The Netherlands, and Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
Email: deweger@few.eur.nl

DOI: https://doi.org/10.1090/S0025-5718-98-00900-4
Received by editor(s): July 22, 1996
Received by editor(s) in revised form: October 7, 1996
Additional Notes: De Weger’s research was supported by the Netherlands Mathematical Research Foundation SWON with financial aid from the Netherlands Organization for Scientific Research NWO
Article copyright: © Copyright 1998 American Mathematical Society