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PreprintCC BY-NC-ND 4.0arXiv

A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant

  • Yaoran Yang
  • Gaishi Yang
  • Yutong Zhang
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Abstract

Let p≥7 be a prime with p≡3pmod4 and let χ be the Legendre symbol modulo p. We prove that det[x+χ(j−k)]0≤ j,k≤(p−7)/2=floor(p−2)/32x in ℂ[x], which settles Conjecture~3.4 of Zhi-Wei Sun. The truncated matrix is a corner of Chapman's Legendre-symbol matrix C, and its determinant can be expressed through a handful of entries of C−1 and of C−1one. Those entries are in turn read off from Vsemirnov's cyclotomic factorization of C with the help of Schur's Pfaffian identity.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026proof,
  title = {{A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant}},
  author = {Yaoran Yang and Gaishi Yang and Yutong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.22548},
  eprint = {2606.22548},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.22548},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant
AU  - Yaoran Yang
AU  - Gaishi Yang
AU  - Yutong Zhang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2606.22548
UR  - https://arxiv.org/abs/2606.22548
AB  - Let p≥7 be a prime with p≡3pmod4 and let χ be the Legendre symbol modulo p. We prove that det[x+χ(j−k)]_0≤ j,k≤(p−7)/2=floor(p−2)/3^2x in ℂ[x], which settles Conjecture~3.4 of Zhi-Wei Sun. The truncated matrix is a corner of Chapman's Legendre-symbol matrix C, and its determinant can be expressed through a handful of entries of C^−1 and of C^−1one. Those entries are in turn read off from Vsemirnov's cyclotomic factorization of C with the help of Schur's Pfaffian identity.
ER  -