A Clean Approach to Rational Cubic Residues

In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result essentially states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M L ≡ t 2 − 1 t3 − 9t mod q for some integer t. In this paper we demonstrate that a similar result may be obtained for cubic nonresidues, yielding a cubic character for fixed p that provides an effective means for ascertaining whether or not an arbitrary integer c is a cubic residue modulo p. As an illustration of this technique, we determine whether 1982 is a cubic residue modulo the 131-digit prime p = 1 4 (3 + 5), a question which, for all practical purposes, is impossible to answer with Lehmer’s original criterion.