Two determinant evaluations in Sun's conjectures involving Legendre symbols

TL;DR

This paper proves two determinant evaluations related to Sun's conjectures involving Legendre symbols, using advanced matrix factorizations and cofactor calculations, with results applicable modulo prime p.

Contribution

It provides the first complete proof of two specific Sun conjectures on determinants of matrices of Legendre symbols, employing novel matrix factorizations and algebraic techniques.

Findings

Resolved the p ≡ 1 mod 4 case of Conjecture 4.8(i)

Evaluated determinants modulo p for Conjecture 4.10(i)

Introduced new matrix factorization methods for Legendre symbol matrices

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 resolves Conjecture 4.10(i) for \(p\equiv1\pmod4\) in a stronger form: for any ordered half-system modulo sign and all \(u,v\in\mathbb F_p\), the determinant is evaluated exactly modulo \(p\). This second proof is a two-antidiagonal Vandermonde factorisation, and it gives the asserted square class as an immediate corollary.