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

Gaishi Yang; Yaoran Yang; Yutong Zhang

arxiv: 2606.22548 · v1 · pith:KZELEAF2new · submitted 2026-06-21 · 🧮 math.NT

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

Yaoran Yang , Gaishi Yang , Yutong Zhang This is my paper

Pith reviewed 2026-06-26 09:46 UTC · model grok-4.3

classification 🧮 math.NT

keywords Legendre symboldeterminanttruncated matrixSun conjectureChapman matrixVsemirnov factorizationSchur-Pfaffian identityprime congruent 3 mod 4

0 comments

The pith

For every prime p congruent to 3 modulo 4, the truncated Legendre-symbol determinant equals floor((p-2)/3) squared times x.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves Conjecture 3.4 of Zhi-Wei Sun by supplying an explicit evaluation of a truncated determinant whose entries are Legendre symbols. For primes p congruent to 3 modulo 4 the determinant equals the square of floor of (p minus 2) divided by 3, multiplied by x. The argument first reduces the truncated determinant to certain inverse entries drawn from Chapman's complete Legendre-symbol matrix. Those inverse entries are then computed by combining Vsemirnov's factorization with a Schur-Pfaffian resolvent identity. A sympathetic reader cares because the result converts an open conjecture into a closed-form statement that links quadratic-residue matrices to concrete arithmetic expressions.

Core claim

The paper establishes that for every prime p ≡ 3 (mod 4) the determinant of the truncated Legendre-symbol matrix equals ⌊(p−2)/3⌋² x. This evaluation is obtained by first expressing the truncated determinant in terms of inverse data from Chapman's full Legendre-symbol matrix and then applying Vsemirnov's factorization along with the Schur-Pfaffian resolvent identity to determine the required inverse entries.

What carries the argument

The reduction of the truncated determinant to inverse data for Chapman's full Legendre-symbol matrix, evaluated using Vsemirnov's factorization and the Schur-Pfaffian resolvent identity.

If this is right

Conjecture 3.4 of Sun holds for all primes p congruent to 3 modulo 4.
The determinant now possesses an explicit closed-form expression in terms of p.
The same reduction technique connects truncated versions to known properties of the full matrix.
The evaluation supplies a concrete link between Legendre-symbol determinants and arithmetic functions of p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

One could test whether a parallel formula exists when p is congruent to 1 modulo 4.
The floor function appearing in the formula may reflect the distribution of quadratic residues in residue classes modulo p.
The method might extend to other truncations or to matrices built from related multiplicative characters.

Load-bearing premise

The reduction relating the truncated determinant to the inverse data of the full Chapman matrix is correct and Vsemirnov's factorization together with the Schur-Pfaffian identity accurately evaluate those data.

What would settle it

Direct numerical computation of the truncated determinant for a small prime such as p=7, where floor((7-2)/3) equals 1, and checking whether the determinant equals x.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves Sun's conjecture on the value of a truncated Legendre-symbol determinant for primes p ≡ 3 mod 4 by reducing it to inverse data on Chapman's full matrix and evaluating via Vsemirnov factorization plus the Schur-Pfaffian identity.

read the letter

The main takeaway is that the authors have closed Conjecture 3.4 of Zhi-Wei Sun. They show the truncated determinant equals floor((p-2)/3)^2 x for those primes.

They do this by linking the truncated matrix to the inverse entries of Chapman's complete Legendre-symbol matrix, then applying Vsemirnov's factorization and the Schur-Pfaffian resolvent identity to compute those entries. The result is a clean evaluation rather than a new general method.

What stands out is that the conjecture is now settled with an explicit formula. The reduction step and the two cited tools are standard in this corner of the literature, so the proof rests on combining them correctly. The authors note the initial draft came from ChatGPT but was human-checked, which at least flags the need for careful reading of the identity applications.

The soft spot is that the argument depends on the reduction to Chapman's inverse data being exact and on the factorization and resolvent applying without hidden restrictions for these specific matrices. The abstract sketches the steps, but any gap in verifying the matrix truncation or the precise invocation of the identities would affect the claim. No internal contradiction shows up from the description, and the cited results are external and previously established.

This is for readers already following Sun's conjectures or working on Legendre-symbol matrices and their determinants. Someone outside that niche gets little beyond the statement that one more case is resolved.

It is worth sending to peer review. The claim is narrow but concrete, the tools are appropriate, and a referee can check the reduction and identity steps in detail.

Referee Report

0 major / 2 minor

Summary. The manuscript proves Conjecture 3.4 of Zhi-Wei Sun by evaluating, for every prime p ≡ 3 mod 4, a truncated Legendre-symbol determinant as ⌊(p-2)/3⌋² x. The argument reduces the truncated determinant to inverse data for Chapman's full Legendre-symbol matrix and evaluates those data using Vsemirnov's factorization together with a Schur-Pfaffian resolvent identity. The proof was generated by ChatGPT and independently verified by the authors.

Significance. If the reduction and subsequent evaluations hold, the result resolves an explicit conjecture on truncated Legendre-symbol determinants by connecting it to the full-matrix inverses studied by Chapman and the factorizations of Vsemirnov. This supplies a closed-form evaluation in a setting where such determinants have been studied but not previously evaluated in the truncated case.

minor comments (2)

The abstract states the target value as ⌊(p-2)/3⌋²x without defining the factor x; the introduction or statement of the conjecture should make the precise target expression explicit.
The reduction from the truncated determinant to the inverse entries of the full Chapman matrix is described at a high level; a short paragraph or diagram clarifying the precise submatrix or entry selection would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The summary accurately captures the content and approach of the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external cited results

full rationale

The paper reduces the truncated Legendre-symbol determinant to inverse entries of Chapman's full matrix, then applies Vsemirnov factorization and the Schur-Pfaffian resolvent identity to obtain the closed form. These supporting results are external (Chapman, Vsemirnov) and are not derived from or fitted to the target conjecture within the paper. No equation equates the claimed value to a fitted parameter or self-defined quantity by construction, and no load-bearing step collapses to a self-citation chain. The argument is therefore self-contained once the cited external identities are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard properties of Legendre symbols, determinants, and two cited external results rather than new free parameters or invented entities.

axioms (2)

standard math Standard algebraic properties of the Legendre symbol and matrix determinants
Invoked to define the truncated matrix and its determinant.
domain assumption Correctness of Vsemirnov's factorization and the Schur-Pfaffian resolvent identity
Used to evaluate the inverse data after reduction to Chapman's matrix.

pith-pipeline@v0.9.1-grok · 5613 in / 1352 out tokens · 35594 ms · 2026-06-26T09:46:13.000401+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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Acta Arith.115(3), 231–244 (2004) https://doi.org/10.4064/aa115-3-4

Chapman, R.: Determinants of Legendre symbol matrices. Acta Arith.115(3), 231–244 (2004) https://doi.org/10.4064/aa115-3-4

work page doi:10.4064/aa115-3-4 2004
[2]

evil determinant

Vsemirnov, M.: On the evaluation of R. Chapman’s “evil determinant”. Linear Algebra Appl.436(11), 4101–4106 (2012) https://doi.org/10.1016/j.laa.2011.08. 039

work page doi:10.1016/j.laa.2011.08 2012
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evil determinant

Vsemirnov, M.: On R. Chapman’s “evil determinant”: casep≡1 (mod 4). Acta Arith.159(4), 331–344 (2013) https://doi.org/10.4064/aa159-4-3

work page doi:10.4064/aa159-4-3 2013
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2024.doi: 10.48550/arXiv.2405

Sun, Z.-W.: Problems and results on determinants involving Legendre sym- bols. arXiv:2405.03626v8 [math.NT] (2024). https://doi.org/10.48550/arXiv.2405. 03626

work page doi:10.48550/arxiv.2405 2024
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work page doi:10.1007/978-1-4757-2103-4 1990

[1] [1]

Acta Arith.115(3), 231–244 (2004) https://doi.org/10.4064/aa115-3-4

Chapman, R.: Determinants of Legendre symbol matrices. Acta Arith.115(3), 231–244 (2004) https://doi.org/10.4064/aa115-3-4

work page doi:10.4064/aa115-3-4 2004

[2] [2]

evil determinant

Vsemirnov, M.: On the evaluation of R. Chapman’s “evil determinant”. Linear Algebra Appl.436(11), 4101–4106 (2012) https://doi.org/10.1016/j.laa.2011.08. 039

work page doi:10.1016/j.laa.2011.08 2012

[3] [3]

evil determinant

Vsemirnov, M.: On R. Chapman’s “evil determinant”: casep≡1 (mod 4). Acta Arith.159(4), 331–344 (2013) https://doi.org/10.4064/aa159-4-3

work page doi:10.4064/aa159-4-3 2013

[4] [4]

2024.doi: 10.48550/arXiv.2405

Sun, Z.-W.: Problems and results on determinants involving Legendre sym- bols. arXiv:2405.03626v8 [math.NT] (2024). https://doi.org/10.48550/arXiv.2405. 03626

work page doi:10.48550/arxiv.2405 2024

[5] [5]

Schur, I.: ¨Uber die darstellung der symmetrischen und der alternierenden gruppe durch gebrochene lineare substitutionen. J. Reine Angew. Math.139, 155–250 (1911) https://doi.org/10.1515/crll.1911.139.155

work page doi:10.1515/crll.1911.139.155 1911

[6] [6]

, TITLE =

Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990). https: //doi.org/10.1007/978-1-4757-2103-4 13

work page doi:10.1007/978-1-4757-2103-4 1990