On the Diophantine equation $|ax^n-by^n|=1$
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- by Michael A. Bennett and Benjamin M. M. de Weger PDF
- Math. Comp. 67 (1998), 413-438 Request permission
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.References
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Additional Information
- Michael A. Bennett
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 339361
- 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
- Received by editor(s): July 22, 1996
- Received by editor(s) in revised form: October 7, 1996
- Additional Notes: De Wegerâs research was supported by the Netherlands Mathematical Research Foundation SWON with financial aid from the Netherlands Organization for Scientific Research NWO
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp. 67 (1998), 413-438
- MSC (1991): Primary 11D41; Secondary 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-98-00900-4
- MathSciNet review: 1434936