I found something fascinating, the number of incongruent quadradic residues of a prime p>2 divides

(p-1). The number of cubic residues of a prime p divides (p-1). The number of quartic residues of a prime p divides (p-1). This continues indefinitely.

I proved this like this, the set of all nth residues of a prime p, forms a group under multiplication modulo p (I showed an inverse exits by the Pigeonhole Principle)-furthermore this is a cyclic group-and by Lagrange's Theorem the order of a subgroup (finite) divides the order of the group (finite). Thus L(n) divides (p-1) where L(n) is the number of incongruent n'th residues.