Some new residuacity criteria

Richard Howard Hudson and Kenneth S. Williams

Vol. 91 (1980), No. 1, 135–143

DOI: 10.2140/pjm.1980.91.135

Abstract

Let e and k be integers ≥ 2 with e odd and k even. Set 2l = L. C. M. (e,k) and let p be a prime with p ≡ 1 (mod 2l) having g as a primitive root. It is shown that the index of e (with respect to g) modulo k can be computed in terms of the cyclotomic numbers of order l. By applying this result with e = 3, k = 4; e = 5, k = 4; e = 3, k = 8; new criteria are obtained for 3 and 5 to be fourth powers (mod p) and for 3 to be an eighth power (mod p).

Mathematical Subject Classification

Primary: 10A15, 10A15

Milestones

Received: 26 October 1979

Published: 1 November 1980