In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation${(x-d)}^5+x^5+{(x+d)}^5=z^n,n\ge 2,$ where $d,x,z\in \mathbb{Z}$ and $d=2^a{5}^b$ with $a,b\ge 0$.

中文翻译：

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三个五分之和的完美幂

在本文中，我们确定了算术级数中三个五次方之和的完美幂。更准确地说，我们完全解决了丢番图方程${(X-d)}^5+X^5+{(X+d)}^5=z^n,n\ge 2,$在哪里$d,X,z\in \mathbb{Z}$和$d=2^{\mathrm{一种}}{5}^b$和$\mathrm{一种},b\ge 0$.