A refined modular approach to the Diophantine equation $x^2 y^{2n}=z^3$ - Mathematics > Number TheoryReport as inadecuate




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Abstract: Let $n$ be a positive integer and consider the Diophantine equation ofgeneralized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns$x,y,z$. Using methods of modular forms and Galois representations forapproaching Diophantine equations, we show that for $n \in \{5, 31\}$ there areno solutions to this equation. Combining this with previously known results,this allows a complete description of all solutions to the Diophantine equationabove for $n \leq 10^7$. Finally, we show that there are also no solutions for$n\equiv -1 \pmod{6}$.



Author: Sander R. Dahmen

Source: https://arxiv.org/







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