A Conjecture on Circular Permutations over Finite Fields

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 $q>18888871$. For $a_0=0$, the conclusion holds for all $q>4$. For $a_0\mathrm{\ne }0$, we apply the Chvátal–Erdős theorem to obtain a Hamilton cycle.