A Conjecture on Circular Permutations over Finite Fields

Source 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.