Solving a power sum congruence

We present a general algorithm for solving the congruence $x^{n}+y^{n}\equiv 0 \pmod m$ for integers $m\ge 2$ and determine the number of solutions. We begin with the case of prime power moduli, where unit solutions are classified using primitive roots and Hensel’s lemma. These solutions are shown to lie on certain modular lines. Nonunit solutions are then described, and we prove that mixed solutions do not occur. Finally, the Chinese remainder theorem yields a complete description of the solutions for arbitrary moduli, and several illustrative examples are provided.