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

Abstract

We prove Conjecture~3.4 of Zhi-Wei Sun by evaluating, for every prime $p\equiv3\pmod4$, a truncated Legendre-symbol determinant as $\lfloor(p-2)/3\rfloor^{2}x$. The argument reduces the determinant to inverse data for Chapman's full Legendre-symbol matrix and evaluates those data using Vsemirnov's factorization and a Schur-Pfaffian resolvent identity. OpenAI's ChatGPT produced the proof, which the authors independently checked and confirmed.