The equation $(x-d)^5+x^5+(x+d)^5=y^n$
Volume 198 / 2021
Acta Arithmetica 198 (2021), 387-399 MSC: 11D41, 11D61. DOI: 10.4064/aa200617-11-11 Published online: 22 February 2021
We solve the equation of the title under the assumption that $\gcd (x,d)=1$ and $n\geq 2$. This generalizes earlier work of the first author, Patel and Siksek (2016). Our main tools include Frey–Hellegouarch curves and associated modular forms, and an assortment of Chabauty-type techniques for determining rational points on curves of small positive genus.