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arXiv · 2606.22548

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 $p\ge7$ be a prime with $p\equiv3\pmod4$ and let $\chi$ be the Legendre symbol modulo $p$. We prove that $\det[x+\chi(j-k)]_{0\le j,k\le(p-7)/2}=\floor{(p-2)/3}^{2}x$ in $\mathbb{C}[x]$, 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 $C^{-1}\one$. 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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Yaoran Yang, Gaishi Yang, Yutong Zhang. 2026-06-21. A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant. https://arxiv.org/abs/2606.22548

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