A Conjecture on Circular Permutations over Finite Fields

Abstract: Let (\mathbb{F}q) be a finite field with (q>7). Zhi-Wei Sun conjectured that for every (a_0\in\mathbb{F}_q), there is a circular permutation ((a_1,\dots,a{q-1})) of non-zero elements of (\mathbb{F}q) such that (a_0+a_i a{i+1}) is primitive for (1\le i\le q-1), where (a_q=a_1). In this paper, we confirm this conjecture for (q>18\,888\,871). For (a_0=0), the conclusion holds for all (q>4). For (a_0\ne0), we apply the Chvátal--Erdős theorem to obtain a Hamilton cycle.