  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.
