On Golomb's conjecture

Abstract

Let F_q be a finite field with q elements and let $\text{F}{}_{\mathrm{q}}{}^{\mathrm{\ast }}$ be the multiplicative group of F_q. In this paper we shall prove

Theorem. If q − 1 = q₁^r_1…q_m^r_m, q⩾2^4m, in particular q>1.16×10¹⁸, then for any integer s with s>1 and every $\text{βϵF}{}_{\mathrm{q}}{}^{\mathrm{\ast }}$, $\text{α}{}_{\mathrm{i}}\text{ϵF}{}_{\mathrm{q}}{}^{\mathrm{\ast }}$ (i=1, …, s), there exist s primitive elements x₁, …, x_s of F_q, where α₁x₁ + … + α_sx_s = β.