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Zhi-Wei Sun’s circular permutation conjecture over finite fields

For every prime power q>7q>7 and every a0∈Fqa_0\in\mathbb{F}_q, there exists a circular permutation (a1,…,aq−1)(a_1,\ldots,a_{q-1}) of Fq×\mathbb{F}_q^{\times} such that, with aq=a1a_q=a_1, each element a0+aiai+1a_0+a_i a_{i+1} is primitive in Fq×\mathbb{F}_q^{\times}; equivalently, ord⁡(a0+aiai+1)=q−1\operatorname{ord}(a_0+a_i a_{i+1})=q-1 for every 1≤i≤q−11\le i\le q-1.

References

Primary source

arXiv

Additional references

extremal-graph-theoryfinite-fields
Posted · edited

Progress summary

Refreshed
Claimed progress

A new unrefereed preprint proves the conjecture for sufficiently large fields, but the full claim in all permitted fields remains open.

Zhi-Wei Sun posed the conjecture on 17 September 2013. For every finite field Fq\mathbb{F}_q with q>7q>7 and every a0∈Fqa_0\in\mathbb{F}_q, it asks for a circular ordering of the nonzero elements whose consecutive products, after adding a0a_0, are all primitive.

Known results

  • Sun, 2013: recorded the statement as an open conjecture; no general proof or counterexample was given.

October 5, 2026 development

Xin-Qi Luo and Yue-Feng She’s unrefereed preprint gives a Hamilton-cycle argument proving the conjecture above an explicit field-size threshold and separately treating a0=0a_0=0 for q>4q>4. Thus only a finite range of field sizes remains for the full conjecture, which the source does not settle.

Current status (as of October 2026): The conjecture is claimed for fields above an explicit threshold and for a0=0a_0=0 when q>4q>4, but the full case q>7q>7 remains open.

Sources

Solutions 0

No solutions have been posted yet.