On the equation $a^{3} + b^{3n} = c^{2}$
Volume 163 / 2014
Acta Arithmetica 163 (2014), 327-343
MSC: Primary 11D41; Secondary 11D61, 11G05, 14G05.
DOI: 10.4064/aa163-4-3
Abstract
We study coprime integer solutions to the equation $a^3 + b^{3n} = c^2$ using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from $\mathbb Q$-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.
