Vol. 16, No. 1, 1966

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ISSN: 0030-8730
The distribution of cubic and quintic non-residues

James Henry Jordan

Vol. 16 (1966), No. 1, 77–85
Abstract

For a prime p 1 (mod 3), the reduced residue system S3, modulo p, has a proper multiplicative subgroup, C0, called the cubic residues modulo p. The other two cosets formed with respect to C0, say C1 and C2, are called classes of cubic non-residues. Similarly for a prime p 1 (mod 5) the reduced residue system S5, modulo p, has a proper multiplicative subgroup, Q0, called the quintic residues modulo p. The other four cosets formed with respect to Q0, say Q1, Q2, Q3 and Q4 are called classes of quintic non-residues. Two functions, f3(p) and f5(p), are sought so that (i) if p 1 (mod 3) then there are positive integers ai Ci, i = 1,2, such that ai < f3(p), and (ii) if p 1 (mod 5) then there are positive integers ai Qi, i = 1,2,3,4 such that ai < f5(p). The results established in this paper are that for p sufficiently large, (i) f3(p) = pα+𝜖, where α is approximately .191, and (ii) f5(p) = pβ+𝜖, where .27 < β < .2725.

Mathematical Subject Classification
Primary: 10.06
Secondary: 10.42
Milestones
Received: 8 April 1963
Revised: 4 May 1964
Published: 1 January 1966
Authors
James Henry Jordan