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A Conjecture on Circular Permutations over Finite Fields
Abstract
Let 𝔽q be a finite field with q>7. Zhi-Wei Sun conjectured that for every a0∈𝔽q, there is a circular permutation (a1,…,aq−1) of non-zero elements of 𝔽q such that a0+ai ai+1 is primitive for 1≤ i≤ q−1, where aq=a1. In this paper, we confirm this conjecture for q>18 888 871. For a0=0, the conclusion holds for all q>4. For a0≠0, we apply the Chvátal--Erdős theorem to obtain a Hamilton cycle.
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BibTeX
@misc{luo2026conjecture,
title = {{A Conjecture on Circular Permutations over Finite Fields}},
author = {Xin-Qi Luo and Yue-Feng She},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06289},
eprint = {2610.06289},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06289},
}
RIS
TY - UNPB
TI - A Conjecture on Circular Permutations over Finite Fields
AU - Xin-Qi Luo
AU - Yue-Feng She
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06289
UR - https://arxiv.org/abs/2610.06289
AB - Let 𝔽_q be a finite field with q>7. Zhi-Wei Sun conjectured that for every a_0∈𝔽_q, there is a circular permutation (a_1,…,a_q−1) of non-zero elements of 𝔽_q such that a_0+a_i a_i+1 is primitive for 1≤ i≤ 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≠0, we apply the Chvátal--Erdős theorem to obtain a Hamilton cycle.
ER -