A refined modular approach to the diophantine equation x² + y²ⁿ = z³

Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x² + y²ⁿ = z³ 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 ∈ {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 ≤ 10⁷. Finally, we show that there are also no solutions for n ≡-1 (mod 6).