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Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places and let $\mu$, $\alpha_1,\alpha_2,\alpha_3$ be non--zero elements in $K$. Denote by $\OS$ the ring of $S$--integers in $K$ and by $\OS^\times$ the group of $S$--units. Then the set of equivalence classes (namely, up to multiplication by $S$--units) of the solutions $(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon)\in\OS^3\times(\OS^\times)^4$ of the diophantine equation $$ (X-\alpha_1 E_1 Y) (X-\alpha_2E_2 Y) (X-\alpha_3E_3 Y)Z=\mu E, $$ satisfying $\Card\{\alpha_1\varepsilon_1,\alpha_2\varepsilon_2,\alpha_3\varepsilon_3\}= 3$, is finite. With the help of this last result, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt's subspace theorem.

Author

Levesque, Claude; Waldschmidt, Michel

Published

2013

Language

EN