Abstract

Let $n$ be a positive integer and consider the Diophantine equation of generalized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns $x,y,z$. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for $n \in \{5, 31\}$ there are no solutions to this equation. Combining
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