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- W2046339960 abstract "In this paper we prove that for $$p > 13649$$ equations of the form $$x^{13} + y^{13} = Cz^{p}$$ have no non-trivial primitive solutions $$(a,b,c)$$ such that $$13 not mid c$$ for an infinite family of values for $$C$$ . Our method consists on relating a solution $$(a,b,c)$$ to the previous equation to a solution $$(a,b,c_1)$$ of another Diophantine equation with coefficients in $$mathbb Q (sqrt{13})$$ . Then we attach to $$(a,b,c_1)$$ a Frey curve $$E_{(a,b)}$$ defined over $$mathbb Q (sqrt{13})$$ that is not a $$mathbb Q $$ -curve. We prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at $$3$$ . This theorem, in particular, implies modularity of $$E_{(a,b)}$$ . This enables us to use level lowering results and apply the modular approach via Hilbert cuspforms over $$mathbb Q (sqrt{13})$$ to prove the non-existence of $$(a,b,c_1)$$ and, consequently, of $$(a,b,c)$$ ." @default.
- W2046339960 created "2016-06-24" @default.
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- W2046339960 date "2013-03-29" @default.
- W2046339960 modified "2023-10-17" @default.
- W2046339960 title "Fermat-type equations of signature $$(13,13,p)$$ via Hilbert cuspforms" @default.
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- W2046339960 doi "https://doi.org/10.1007/s00208-013-0920-7" @default.
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