# The generalized Fermat equation with exponents 2, 3, $n$

@article{Freitas2019TheGF,
title={The generalized Fermat equation with exponents 2, 3, \$n\$},
author={Nuno Freitas and Bartosz Naskręcki and Michael Stoll},
journal={Compositio Mathematica},
year={2019},
volume={156},
pages={77 - 113}
}
• Published 15 March 2017
• Mathematics
• Compositio Mathematica
We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$ , to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$ . We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti…
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