Two determinant evaluations in Sun's conjectures involving Legendre symbols
Pith reviewed 2026-05-20 02:30 UTC · model grok-4.3
The pith
Determinants of Legendre symbol matrices equal leg(2/p) times the difference of two summed quadratic residues.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper proves that det H equals leg(2/p)(b'_p - a'_p) together with the explicit 2x2 package U^T H^{-1} U for the adjacent minor of the Legendre matrix, where U equals the vector of ones and eta_j equals chi(j). It further proves that for any ordered half-system modulo sign and all u, v in the field F_p the determinant equals a specific value modulo p. Both proofs rely on extending Vsemirnov's factorization of Chapman's matrix and the two-antidiagonal Vandermonde factorization to the Legendre-symbol setting.
What carries the argument
Vsemirnov's factorization of Chapman's matrix with adjacent cofactor calculation for the first determinant, and the two-antidiagonal Vandermonde factorization applied directly to the Legendre-symbol matrix over any ordered half-system for the second result.
If this is right
- The p congruent to 1 mod 4 part of Conjecture 4.8(i) is resolved by the determinant formula and the inverse package.
- Conjecture 4.10(i) is resolved in a stronger form that holds for every ordered half-system modulo sign.
- The asserted square class in the second conjecture follows at once from the modular determinant evaluation.
- The explicit inverse allows direct solution of linear systems whose coefficient matrix is the adjacent minor of the Legendre matrix.
Where Pith is reading between the lines
- The same factorization approach could be tested on matrices built from other quadratic characters or from higher-order characters.
- The modular evaluation for arbitrary half-systems suggests that similar exact formulas may exist when the half-system is replaced by other subsets closed under negation.
- Numerical checks for small primes outside the p congruent to 1 mod 4 range could reveal whether analogous closed forms hold or break in a controlled way.
Load-bearing premise
Vsemirnov's factorization and the two-antidiagonal Vandermonde identity extend without modification to the specific Legendre-symbol matrices and half-systems considered here.
What would settle it
A direct computation for a small prime p congruent to 1 mod 4, such as p=13, showing that the determinant of the constructed matrix H differs from leg(2/p) times (b'_p - a'_p) would disprove the first evaluation.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. For the first, it resolves the p≡1 mod 4 part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to the explicit four-entry inverse package U^T H^{-1} U for the adjacent minor [χ(j-k+1)]_{0≤j,k<(p-1)/2}, yielding det H = leg(2/p)(b'_p - a'_p) together with the displayed 2×2 matrix involving U=(1,η) where η_j=χ(j). The proof relies on Vsemirnov's factorization of Chapman's matrix combined with an adjacent cofactor calculation. For the second, it resolves Conjecture 4.10(i) for p≡1 mod 4 in a stronger form: an exact evaluation modulo p of the determinant for any ordered half-system modulo sign and all u,v in F_p, obtained via a two-antidiagonal Vandermonde factorization, with the square-class corollary as an immediate consequence.
Significance. If the derivations hold, the results supply explicit closed-form evaluations for determinants of Legendre-symbol matrices that directly settle specific cases of Sun's conjectures. The combination of Vsemirnov's prior factorization with a targeted cofactor computation for the first claim, and the Vandermonde factorization for the second, provides a clean algebraic route without fitted parameters or self-referential definitions. The stronger mod-p statement for arbitrary ordered half-systems is a notable strengthening. The stress-test concern about direct extension of the factorizations does not land on the manuscript: the adjacent cofactor calculation explicitly adapts Vsemirnov's factorization to the Legendre entries and fixed half-system, while the Vandermonde step is applied to the signed ordering with parameters u,v, yielding the claimed exact evaluations.
minor comments (2)
- [core evaluation] In the displayed formula for U^T H^{-1} U, the (1,1) entry contains the ratio (p b'_p - a'_p)/(b'_p - a'_p); a brief remark on why the factor p appears (arising from the cofactor or the underlying matrix scaling) would aid readability.
- [introduction] The abstract refers to 'four indeterminates' in the first conjecture but the stated evaluation is for det H; a one-sentence clarification in the introduction linking the indeterminates to the reduction step would prevent any momentary confusion.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and the positive recommendation to accept. The referee's summary correctly identifies the two main results and their relation to Sun's conjectures, including the strengthening in the second evaluation.
read point-by-point responses
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Referee: No major comments are listed; the report focuses on the summary, significance, and recommendation to accept, with a note that the stress-test concern does not apply to the manuscript.
Authors: We appreciate the referee's confirmation that the adjacent cofactor calculation adapts Vsemirnov's factorization appropriately to the Legendre-symbol setting and that the Vandermonde factorization yields the claimed exact evaluations. No revisions are required in response to the report. revision: no
Circularity Check
No circularity; derivations rest on independent external factorizations
full rationale
The paper reduces the target determinants to Vsemirnov's factorization of Chapman's matrix (for the first result) together with an adjacent cofactor calculation, and to the standard two-antidiagonal Vandermonde factorization (for the second result). Both are cited as prior, independent results whose validity does not depend on the Legendre-symbol matrices or the Sun conjectures under consideration. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the stated proof chain. The evaluations are therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Legendre symbol satisfies the usual quadratic reciprocity and multiplicativity properties over F_p.
- domain assumption Vsemirnov's factorization of Chapman's matrix applies to the adjacent minor of the Legendre matrix.
discussion (0)
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