Problem

Source: 2019 Dürer Math Competition Finals Day2 E+15 https://artofproblemsolving.com/community/c1621835_2019_

Tags: diophantine, Diophantine equation, number theory, Sum



The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$How many possibilities are there for the value of $m$?