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Let $x, y, z$ be pairwise coprime positive integers. Does one have $x^5 + y^5 = z^p$ for any prime $p \geq 2$ ?

Let $x, y, z$ be positive integers. Does one have $x^5 + y^5 = z^p$ for any prime $p \geq 2$ ?

Let $x, y, z$ be pairwise coprime positive integers. Does one have $x^5 + y^5 = z^p$ for any prime $p \geq 2$ ?

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On the Diophantine equation $x^{5} + y^5 = z^p$

Let $x, y, z$ be positive integers. Does one have $x^5 + y^5 = z^p$ for any prime $p \geq 2$ ?