Proofs of some geometric conjectures on the power sum congruence modulo a prime

The main purpose of this paper is to verify the geometric conjectures of Mustonen (2022) concerning the solutions and the number of solutions of the congruence xⁿ + yⁿ ≡ 0 (mod p), where p is a prime. For p>2, the nontrivial solutions lie on the “lines” y ≡ cx (mod p), where c ranges over the n-th roots of -1 modulo p. The total number of solutions is 1+(p-1)d if d divides (p-1)/2, and 0 otherwise, where d=gcd(n, p-1). For each c, the lines are equally spaced.