The generalized Fermat equation with exponents 2, 3, n

Authors:

• Nuno Freitas,
• Bartosz Naskręcki,
• Michael Stoll

Abstract

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-symplectically isomorphic$p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$that have to be considered, based on local information at 2 and 3; this list depends on$p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when$p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case$p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.