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

A refined modular approach to the Diophantine equation $x^2 y^{2n}=z^3$ - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

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/