Two determinant evaluations in Sun's conjectures involving Legendre symbols

Abstract

We prove two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. The first one resolves the \(p\equiv1\pmod4\) part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor \([\chi(j-k+1)]_{0\le j,k<(p-1)/2}\). The core evaluation is \[ \det H=\leg{2}{p}(b'_p-a'_p),\qquad U^TH^{-1}U= \begin{pmatrix} \leg{2}{p}\dfrac{pb'_p-a'_p}{b'_p-a'_p}&1\\[2mm] \dfrac{b'_p-a'_p-1}{b'_p-a'_p}&1 \end{pmatrix}, \] where \(U=(\mathbf1,\eta)\) and \(\eta_j=\chi(j)\). The proof uses Vsemirnov's factorisation of Chapman's matrix and an adjacent cofactor calculation. The second result gives a uniform exact congruence modulo \(p\) for the determinant underlying Sun's Conjecture 4.10(i), valid for any ordered half-system modulo sign and all \(u,v\in\mathbb F_p\). Its standard specialization recovers the asserted square class. The square-class assertion itself also follows from Sun's earlier evaluation of \(T(d,p)\); the contribution here is an exact and half-system refinement.