AI 中文总结
本文证明了足够大阶数的素数圆存在，并研究了有限域上本原和差图及有向图的哈密顿性，给出了素数幂和素数域上的界限。
AI 中文摘要
一个阶为$2n$的素数圆是将$1,\ldots,2n$排成一个圆排列，使得任意两个相邻项之和为素数。我们证明了对于每一个足够大的$n$，素数圆都存在。证明基于完美匹配和鲁棒扩张。我们还研究了由有限域上的本原和与本原差定义的图和有向图中的哈密顿圈。特别地，对于每个素数幂$q>18\\,888\\,871$，$\n\F_q$上的本原和图是哈密顿的；而对于完整素数域$\F_p$上的图，该界限改进为$p>61$。
英文摘要
A prime circle of order $2n$ is a circular ordering of $1,\ldots,2n$ such that the sum of every two adjacent terms is prime. We prove that a prime circle exists for every sufficiently large $n$. The proof is based on a perfect matching and robust expansion. We also study Hamilton cycles in graphs and digraphs defined by primitive sums and differences over finite fields. In particular, the primitive-sum graph on $\F_q$ is Hamiltonian for every prime power $q>18\,888\,871$, and for the graph on a full prime field $\F_p$, the bound improves to $p>61$.
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