English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 equivalenceclasses (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 thediophantine equation $$ (X-\alpha_1 E_1 Y) (X-\alpha_2E_2 Y) (X-\alpha_3E_3Y)Z=\mu E, $$ 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 ofThue-Mahler equations having only trivial solutions. Furthermore, we produce aneffective upper bound for the number of these solutions. The proofs of thispaper rest heavily on Schmidt's subspace theorem.

Familles d'equations de Thue-Mahler n'ayant que des solutions triviales