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

Bugeaud, Yann ; Mignotte, Maurice

Canadian journal of mathematics, Tome 60 (2008) no. 3, pp. 491-519

Résumé

We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation $${{5}^{u}}{{x}^{n}}-{{2}^{r}}{{3}^{5}}{{y}^{n}}=\pm 1,$$ in non-zero integers $x,y$ and positive integers $u,r,s$ and $n\ge 3$ . 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.

Détail

DOI : 10.4153/CJM-2008-024-9

Mots-clés : 11F80, 11D61, 11D59, 11J86, 11Y50, Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equations

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