We study coprime integer solutions to the equation a³ + b³ⁿ = c² 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 ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-3,
author = {Michael A. Bennett and Imin Chen and Sander R. Dahmen and Soroosh Yazdani},
title = {On the equation a3 + b3n = c2},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {327-343},
zbl = {1306.11025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-3}
}
Michael A. Bennett; Imin Chen; Sander R. Dahmen; Soroosh Yazdani. On the equation a³ + b³ⁿ = c². Acta Arithmetica, Tome 166 (2014) pp. 327-343. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-3/