Question 86
Let $p$ be a prime number and $n_1, n_2, \dots, n_k \in \{1, 2, \dots, p-1\}$ be positive integers. Show that the equation
$$
x_1^{n_1} + x_2^{n_2} + \dots + x_k^{n_k} = x_{k+1}^p
$$
has infinitely many positive integer solutions.
$$
x_1^{n_1} + x_2^{n_2} + \dots + x_k^{n_k} = x_{k+1}^p
$$
has infinitely many positive integer solutions.