A Clean Approach to Rational Cubic Residues

Details A Clean Approach to Rational Cubic Residues A Clean Approach to Rational Cubic Residues

2006-11-06

arxiv.org/abs/math/0611151v2

Mathematics

Number Theory

15 pages, submitted to International Journal of Number Theory, one paragraph appended to section five in v2

Scientific paper

In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. 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=(3^19+5^82)/4, a question which is essentially impossible to answer with Lehmer's original criterion.

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