# 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} }

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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Let P greater-than-or-equal-to 3 and E prime slash double-struck upper Q be elliptic curves with isomorphic <inline-formula content-type="math/mathml" xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext) be a prime.

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In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^\alpha C^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for…

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