Two determinant evaluations in Sun's conjectures involving Legendre symbols

Abstract

We prove two determinant evaluations attached to Sun&#39;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 \([χ(j-k+1)]_{0\le j,k<(p-1)/2}\). The core evaluation is \[ \det H=\leg{2}{p}(b&#39;_p-a&#39;_p),\qquad U^TH^{-1}U= \begin{pmatrix} \leg{2}{p}\dfrac{pb&#39;_p-a&#39;_p}{b&#39;_p-a&#39;_p}&1\\[2mm] \dfrac{b&#39;_p-a&#39;_p-1}{b&#39;_p-a&#39;_p}&1 \end{pmatrix}, \] where \(U=(\mathbf1,η)\) and \(η_j=χ(j)\). The proof uses Vsemirnov&#39;s factorisation of Chapman&#39;s matrix and an adjacent cofactor calculation. The second result gives a uniform exact congruence modulo \(p\) for the determinant underlying Sun&#39;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&#39;s earlier evaluation of \(T(d,p)\); the contribution here is an exact and half-system refinement.