The distribution of cubic and quintic non-residues

James Henry Jordan

Vol. 16 (1966), No. 1, 77–85

DOI: 10.2140/pjm.1966.16.77

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 a_i ∈ Cⁱ, i = 1,2, such that a_i < f₃(p), and (ii) if p ≡ 1 (mod 5) then there are positive integers a_i ∈ Qⁱ, i = 1,2,3,4 such that a_i < f₅(p). The results established in this paper are that for p sufficiently large, (i) f₃(p) = p^α+𝜖, where α is approximately .191, and (ii) f₅(p) = p^β+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