In this paper we study the equation \(\begin{aligned} (x-dr)^5+\cdots +(x-r)^5+x^5+(x+r)^5+\cdots +(x+dr)^5=y^p \end{aligned}\) under the condition \(\gcd (x,r)=1\) . We present a recipe for proving the non-existence of non-trivial integer solutions of the above equation, and as an application we obtain explicit results for the cases \(d=2,3\) (the case \(d=1\) was already solved [Bennett and Koutsianas in Acta Arith 198(4):387–399, 2021]). We also prove an asymptotic result for \(d\equiv 1, 7\pmod 9\) . Our main tools include the modular method and employing Frey curves and their associated modular forms, as well as the symplectic argument.
