The distribution of cubic and quintic non-residues

For a prime p = 1 (mod 3), the reduced residue system S3, modulo p, has a proper multiplicative subgroup, C°, called the cubic residues modulo p. The other two cosets formed with respect to G°, say C1 and C2, are called classes of cubic non-residues. Similarly for a prime J J Ξ I (mod5) the reduced residue system S5, modulo p, has a proper multiplicative sub-group, Q°, called the quintic residues modulo p. The other four cosets formed with respect to Q°, say Q1, Q2, Q3 and Q4 are called classes of quintic non-residues. Two functions, fs(p) and /5(p), are sought so that (i) if p = 1 (mod 3) then there are positive integers di^C1, i = 1, 2, such that aι < /3(p), and (ii) if p Ξ l (mod5) then there are positive integers α;eQ% i = 1, 2, 3, 4 such that α; < /5(p). The results established in this paper are that for p sufficiently large, (i) fs(p) = pa+s, where a is approximately.191, and (ii) /5(p) = pβ+*, where.27 < β <.2725. Davenport and Erdos [3] raised the general question about the size of the smallest element in any given class of kth power non-residues. The special cases k = 3 and k = 5 are of primary concern in this paper. They proved a quite general theorem of which two special cases are: THEOREM A. For sufficiently large primes p = 1 (mod 3) and e> 0 each class of cubic non-residues possess a positive integer smaller than p™ι™+\ THEOREM B. For sufficiently large primes p = 1 (mod 5) and ε> 0 each class of quintic non-residues possess a positive integer smaller than pmi396+\ In the same paper Davenport and Erdos used a result of de Bruijn [2] to improve the constant of Theorem A to approximately.383. Recently D. A. Burgess [1] succeeded in improving Polya's in-equality concerning character sums. Burgess ' result is THEOREM C. If p is a prime and if χ is a nonprincipal charac-ter, modulo p, and if H and r are arbitrary positive integers the