On Golomb's conjecture
Let \({\mathbb{F}}_ q\) denote the finite field of order q and let \({\mathbb{F}}_ q^*\) denote its multiplicative group. In a 1984 paper [J. Comb. Theory, Ser. A 37, 13-21 (1984; Zbl 0547.05020)] \textit{S. W. Golomb} conjectured that for q sufficiently large, every \(\beta \in {\mathbb{F}}_ q^*\) is the sum of two primitive elements of \({\mathbb{F}}_ q\). Recently, \textit{J. Wang} [Sci. Sin., Ser. A 31, No.2, 152-161 (1988; Zbl 0687.12012)] established this conjecture for the special case of \(\beta =1\). The present paper settles Golomb's conjecture by proving the following general result: Theorem. If \(q-1=q_ 1^{r_ 1}q_ 2^{r_ 2}...q_ m^{r_ m}\), \(q\geq 2^{4m}\), in particular \(q>1.16\times 10^{18}\), then for any positive integers s with \(s>1\) and for every \(\beta\) and \(\alpha_ 1\), \(\alpha_ 2,...,\alpha_ s\) in \({\mathbb{F}}^*_ q\), there exist s primitive elements \(x_ 1,x_ 2,...,x_ s\in {\mathbb{F}}^*_ q\) such that \(\alpha_ 1x_ 1+\alpha_ 2x_ 2+...+\alpha_ sx_ s=\beta.\) Reviewer's Comment. While this result settles a general form of Golomb's conjecture, other recent research has centered on establishing such results for all q (greater than some small number of exceptional cases); e.g., see [\textit{S. D. Cohen} and \textit{G. L. Mullen}, Primitive elements in finite fields and Costas arrays, Appl. Algebra Eng. Commun. Comput. 2, No.1, 45-53 (1991)].
scientific article; zbMATH DE number 4191827