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In this paper, 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 [BPS16]. Our main tools include Frey-Hellegouarch curves andassociated modular forms, and an assortment of Chabauty-type techniques fordetermining rational points on curves of small positive genus.

The equation $(x-d)^5 + x^5 + (x+d)^5 = y^n$