On the sum of fifth powers in arithmetic progression

In this paper we study equation $$(x-dr)^5+\cdots+x^5+\cdots+(x+dr)^5=y^p$$ 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). We also prove an asymptotic result for \(d\equiv 1, 7\pmod9\). Our main tools include the modular method, employing Frey curves and their associated modular forms, as well as the symplectic argument.