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# A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

Published:2008-06-01
Printed: Jun 2008
• Yann Bugeaud
• Maurice Mignotte
• Samir Siksek
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## Abstract

We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation $5^u x^n-2^r 3^s y^n= \pm 1,$ in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.
 Keywords: Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equation
 MSC Classifications: 11F80 - Galois representations 11D61 - Exponential equations 11D59 - Thue-Mahler equations 11J86 - Linear forms in logarithms; Baker's method 11Y50 - Computer solution of Diophantine equations

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