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

TL;DR

This paper employs modular forms and Galois representations to prove the nonexistence of solutions for the Diophantine equation $x^2+y^{2n}=z^3$ for specific values of n, extending to a complete solution for n up to 10^7.

Contribution

It introduces a refined modular approach to solve the equation for certain n and completes the classification of solutions up to 10^7, including new nonexistence results.

Findings

No solutions for n=5 and n=31.

Complete solutions classified for n ≤ 10^7.

No solutions for n ≡ -1 mod 6.

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 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 $n\equiv -1 \pmod{6}$.