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 $d\equiv 1,7(\mathrm{m}\mathrm{o}\mathrm{d}9)$. Our main tools include the modular method, employing Frey curves and their associated modular forms, as well as the symplectic argument.