A refined modular approach to the Diophantine equation $x^2+y^{2n}=z^3$

Details A refined modular approach to the Diophantine equation $x^2+y^{2n}=z^3$ A refined modular approach to the Diophantine equation $x^2+y^{2n}=z^3$

2010-01-29

arxiv.org/abs/1002.0020v1

Mathematics

Number Theory

12 pages

Scientific paper

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 this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for $n \leq 10^7$. Finally, we show that there are also no solutions for $n\equiv -1 \pmod{6}$.

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