GOLOMB'S CONJECTURE IS TRUE FOR CERTAIN CLASSES OF FINITE FIELDS

Golomb recently made the following conjecture: In any finite field, GF(p~n), where p is a prime integer, there exist two primitive elements with a sum equal to one, the unit element of the field. The purpose of this note is to establish some special cases of Golomb’s conjecture. To achieve this, the following theorem is used. Theorem 1. Let c=φ(m) and let q,q,…,qbe the different prime divisors