Sun-type determinant and permanent congruences

Yaoran Yang; Yutong Zhang

arxiv: 2605.19502 · v1 · pith:4X2Q3NIGnew · submitted 2026-05-19 · 🧮 math.NT

Sun-type determinant and permanent congruences

Yaoran Yang , Yutong Zhang This is my paper

Pith reviewed 2026-05-20 02:39 UTC · model grok-4.3

classification 🧮 math.NT

keywords Sun conjecturesdeterminant congruencespermanent congruencesbinary quadratic formsquadratic extensionfinite fieldsMorley congruenceCayley transform

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The pith

Sun's conjectures on congruences for determinants and permanents modulo primes are proven, with the determinant results strengthened by a root-quotient criterion on irreducible binary quadratic forms over quadratic extensions of prime-order

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

This paper proves seven specific conjectures from Sun's list on congruences satisfied by determinants and permanents of matrices over residue classes modulo a prime. The determinant side receives a strengthened formulation as a root-quotient criterion that classifies the behavior of determinants attached to irreducible binary quadratic forms over the quadratic extension of the field with p elements. The proofs proceed by combining finite-field diagonalisation, cancellation along Cauchy cycles, matching expansions, polynomial interpolation, Morley's congruence, and inspection of leading nonzero terms. These methods also yield higher-order divisibility statements for certain Cayley-type determinants and both mod-p and mod-p-squared results for permanents and signed determinants. A sympathetic reader would care because the work converts an open list of modular identities into established facts while supplying a new arithmetic criterion that may apply to further matrix constructions.

Core claim

The article proves Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list. The determinant part is strengthened to a root-quotient criterion for irreducible binary quadratic forms over the quadratic extension of the prime field; the criterion gives the stated result for the determinant associated with the remaining binary quadratic form. The Cauchy-kernel part gives both derangement congruences modulo the square of the prime and a polynomial fixed-point permanent congruence modulo the prime. The Cayley-transform part gives the signed fixed-point determinant congruences, the quadratic-residue assertion for the signed derangement determinant, and the full fixed-point p

What carries the argument

The root-quotient criterion for irreducible binary quadratic forms over the quadratic extension of the prime field, which classifies the p-adic valuation and congruence class of the associated determinant.

If this is right

Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 all hold.
The determinant of the matrix attached to any irreducible binary quadratic form satisfies the root-quotient criterion over the quadratic extension of the prime field.
Derangement determinants and permanents satisfy the stated congruences modulo p squared.
Signed fixed-point determinants satisfy both the congruence and the quadratic-residue claims.
The half-size quadratic Cayley determinant is divisible by p squared and, in the stronger congruence class, by p cubed.

Where Pith is reading between the lines

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

The same combination of diagonalisation and cycle-cancellation methods could be tested on the remaining unproved entries in Sun's original list.
The root-quotient criterion may extend to determinants attached to reducible forms or to forms over extensions of higher degree.
Analogous congruence statements might hold when the underlying ring is replaced by the integers of a number field rather than Z/pZ.

Load-bearing premise

The listed techniques of finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence, and first nonzero determinant terms are together sufficient to establish the stated congruences and the strengthened criterion without hidden gaps or unstated restrictions on the prime or the quadratic forms.

What would settle it

A concrete counterexample consisting of a small prime p together with an explicit irreducible binary quadratic form over the quadratic extension of F_p for which the determinant fails to match the value predicted by the root-quotient criterion, or for which one of the seven listed congruences does not hold.

read the original abstract

Sun proposed a collection of congruence and quadratic-residue conjectures for determinants and permanents over residue classes modulo a prime. This article proves Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list. The determinant part is strengthened to a root-quotient criterion for irreducible binary quadratic forms over the quadratic extension of the prime field; the criterion gives the stated result for the determinant associated with the remaining binary quadratic form. The Cauchy-kernel part gives both derangement congruences modulo the square of the prime and a polynomial fixed-point permanent congruence modulo the prime. The Cayley-transform part gives the signed fixed-point determinant congruences, the quadratic-residue assertion for the signed derangement determinant, and the full fixed-point permanent congruence modulo the square of the prime. The half-size quadratic Cayley determinant is treated by a local expansion at a simple zero eigenvalue, giving divisibility by the square of the prime and, in the stronger congruence class, by its cube. The proofs combine finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence and first nonzero determinant terms.

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

The paper proves several Sun conjectures on det and perm congruences with a new root-quotient strengthening, but the local expansion for the half-size Cayley determinant rests on an unverified simplicity assumption for the zero eigenvalue.

read the letter

This paper proves Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list and strengthens the determinant side to a root-quotient criterion for irreducible binary quadratic forms over the quadratic extension of the prime field. The proofs use finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence and first nonzero terms to obtain the stated modular results, including derangement congruences mod p squared and fixed-point permanent congruences mod p and mod p squared. The Cayley-transform sections deliver the signed fixed-point determinant congruences and the quadratic-residue claim for the signed derangement determinant. These are direct algebraic arguments rather than reductions to fitted parameters, which is a clear plus. The work fills in explicit proofs for previously open statements and adds the root-quotient refinement without introducing new free parameters or invented entities. The main soft spot is the local expansion at a simple zero eigenvalue for the half-size quadratic Cayley determinant. The argument claims p squared or p cubed divisibility from that expansion, but if the algebraic multiplicity exceeds one for some irreducible forms or small p, the Taylor order drops and the valuation may fall short. The listed techniques do not automatically guarantee simplicity across all cases, so this needs explicit checking in the full text. The paper is aimed at specialists in algebraic combinatorics and modular arithmetic who already follow Sun-type problems. It supplies concrete resolutions and a modest strengthening, which makes it worth referee time even if the eigenvalue step requires extra justification. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves Conjectures 4.6, 4.7, 4.8(ii), 4.9, 4.10(ii), 4.11 and 4.12 from Sun's list of determinant and permanent congruences modulo a prime. The determinant results are strengthened to a root-quotient criterion for irreducible binary quadratic forms over the quadratic extension of the prime field. Proofs combine finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence and first nonzero determinant terms; the half-size quadratic Cayley determinant is handled by local expansion at a simple zero eigenvalue to obtain p^2 or p^3 divisibility.

Significance. If the claims hold, the work resolves a substantial block of open conjectures on modular properties of determinants and permanents, while the root-quotient strengthening supplies a more general algebraic criterion that may apply to other quadratic forms over finite fields. The direct use of standard finite-field techniques without ad-hoc parameters or fitted constants is a methodological strength.

major comments (1)

[half-size quadratic Cayley determinant] Treatment of the half-size quadratic Cayley determinant (as described in the abstract and the relevant proof section): the local expansion at a simple zero eigenvalue is invoked to obtain the asserted p^2 (or p^3) divisibility and thereby the strengthened root-quotient criterion. No explicit argument is supplied that the algebraic multiplicity remains one for every irreducible binary quadratic form over F_p or F_{p^2} and for all primes in the stated range. If multiplicity exceeds one, the Taylor order increases and the valuation may drop, directly affecting Conjectures 4.8(ii), 4.10(ii) and 4.12.

minor comments (2)

[Introduction] The introduction would benefit from an explicit statement of the root-quotient criterion before the technical sections begin.
A short table or list summarizing which conjecture is proved by which technique would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment on the half-size quadratic Cayley determinant. We address the concern point by point below. The comment correctly identifies that an explicit argument for the simplicity of the zero eigenvalue is not supplied in the current version, and we will revise the manuscript to include one.

read point-by-point responses

Referee: Treatment of the half-size quadratic Cayley determinant (as described in the abstract and the relevant proof section): the local expansion at a simple zero eigenvalue is invoked to obtain the asserted p^2 (or p^3) divisibility and thereby the strengthened root-quotient criterion. No explicit argument is supplied that the algebraic multiplicity remains one for every irreducible binary quadratic form over F_p or F_{p^2} and for all primes in the stated range. If multiplicity exceeds one, the Taylor order increases and the valuation may drop, directly affecting Conjectures 4.8(ii), 4.10(ii) and 4.12.

Authors: We agree with the referee that the current manuscript invokes the local Taylor expansion at a simple zero eigenvalue without supplying an explicit proof that the algebraic multiplicity is one in all required cases. This is a genuine expository gap. In the revised version we will insert a new lemma establishing that, for any irreducible binary quadratic form over F_p or F_{p^2} and for all primes p in the stated range, the zero eigenvalue of the associated half-size Cayley matrix is algebraically simple. The argument will use the irreducibility of the form (equivalently, the discriminant being a nonsquare) to show that the derivative of the characteristic polynomial does not vanish at the relevant point, thereby guaranteeing that the first-order term in the local expansion is nonzero. This will secure the claimed p^2 (and p^3) divisibility and confirm the strengthened root-quotient criterion for the affected conjectures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent algebraic techniques.

full rationale

The paper establishes the listed Sun conjectures through direct applications of finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence, and first nonzero determinant terms, as described in the abstract. The half-size quadratic Cayley determinant treatment via local expansion at a simple zero eigenvalue is presented as a computational step yielding the claimed p^2 or p^3 divisibility without reducing the final congruences or root-quotient criterion to fitted parameters, self-definitions, or load-bearing self-citations. No equations or steps in the provided derivation chain equate outputs to inputs by construction, and the techniques are standard and externally verifiable. This is the expected honest non-finding for a paper whose central claims rest on explicit algebraic manipulations rather than circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard axioms of finite fields, determinants, and congruences; no new free parameters or invented entities are introduced.

axioms (2)

standard math Standard algebraic properties of finite fields, quadratic extensions of prime fields, and matrix diagonalization over them hold.
Invoked for the root-quotient criterion and finite-field diagonalisation steps.
standard math Morley's congruence and related interpolation results apply in the stated modular settings.
Used explicitly in the proof outline for the congruences.

pith-pipeline@v0.9.0 · 5745 in / 1444 out tokens · 51898 ms · 2026-05-20T02:39:14.294693+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proofs combine finite-field diagonalisation, Cauchy cycle cancellation, matching expansions, interpolation, Morley's congruence and first nonzero determinant terms.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 6 ... Dp(a,b) ≡ ∏ (λ^m + μ^m) (mod p) for roots of irreducible quadratic

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Borchardt, C.W.: Bestimmung der symmetrischen Verbindungen vermittelst ihrer erzeugenden Function. J. Reine Angew. Math. 53, 193–198 (1855)

work page
[2]

Acta Arith

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
[3]

Linear Multilinear Algebra 72(7), 1071–1077 (2024)

Guo, X., Li, X., Tao, Z., Wei, T.: The eigenvectors-eigenvalues identity and Sun’s conjectures on determinants and permanents. Linear Multilinear Algebra 72(7), 1071–1077 (2024). https://doi.org/10.1080/03081087.2023.2172380

work page doi:10.1080/03081087.2023.2172380 2024
[4]

Graduate Texts in Mathematics, vol

Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990)

work page 1990
[5]

Electron

Ishikawa, M., Kawamuko, H., Okada, S.: A Pfaffian-Hafnian analogue of Bor- chardt’s identity. Electron. J. Combin. 12, Research Paper N9 (2005)

work page 2005
[6]

Krattenthaler, C.: Advanced determinant calculus. Sem. Lothar. Combin. 42, Article B42q (1999)

work page 1999
[7]

Luo, X.-Q., Sun, Z.-W.: Legendre symbols related to certain determinants. Bull. Malays. Math. Sci. Soc. 46, Article 119 (2023). https://doi.org/10.1007/s40840- 023-01505-2 16

work page doi:10.1007/s40840- 2023
[8]

Indian J

Luo, X.-Q., Xia, W.: Legendre symbols related toD p(b,1). Indian J. Pure Appl. Math. (2025). https://doi.org/10.1007/s13226-025-00882-7

work page doi:10.1007/s13226-025-00882-7 2025
[9]

Encyclopedia of Mathematics and Its Applications, vol

Minc, H.: Permanents. Encyclopedia of Mathematics and Its Applications, vol. 6. Addison-Wesley, Reading (1978)

work page 1978
[10]

Morley, F.: Note on the congruence 2 4n ≡(−1) n2n n (mod (2n+ 1) 3). Am. J. Math. 17(2), 168–170 (1895). https://doi.org/10.2307/2369517

work page doi:10.2307/2369517
[11]

She, Y.-F., Sun, Z.-W.: Some determinants involving binary forms. Bull. Malays. Math. Sci. Soc. 48, Article 132 (2025). https://doi.org/10.1007/s40840-025-01900- x

work page doi:10.1007/s40840-025-01900- 2025
[12]

arXiv:2208.12167 (2022)

She, Y.-F., Sun, Z.-W., Xia, W.: A novel permanent identity with applications. arXiv:2208.12167 (2022). https://doi.org/10.48550/arXiv.2208.12167

work page doi:10.48550/arxiv.2208.12167 2022
[13]

Finite Fields Appl

Sun, Z.-W.: On some determinants with Legendre symbol entries. Finite Fields Appl. 56, 285–307 (2019). https://doi.org/10.1016/j.ffa.2018.12.004

work page doi:10.1016/j.ffa.2018.12.004 2019
[14]

Sun, ‘On some determinants and permanents’,Acta Math

Sun, Z.-W.: On some determinants and permanents. Acta Math. Sinica Chin. Ser. 67(2), 286–295 (2024). https://doi.org/10.12386/A20220195

work page doi:10.12386/a20220195 2024
[15]

Electron

Wang, H., Sun, Z.-W.: Proof of a conjecture involving derangements and roots of unity. Electron. J. Combin. 30(2), Article P2.1 (2023). https://doi.org/10.37236/11377

work page doi:10.37236/11377 2023
[16]

Wang, H., Sun, Z.-W.: On certain determinants and related Legen- dre symbols. Bull. Malays. Math. Sci. Soc. 47, Article 58 (2024). https://doi.org/10.1007/s40840-024-01650-2

work page doi:10.1007/s40840-024-01650-2 2024
[17]

Wu, H.-L., She, Y.-F., Ni, H.-X.: A conjecture of Zhi-Wei Sun on determi- nants over finite fields. Bull. Malays. Math. Sci. Soc. 45, 2405–2412 (2022). https://doi.org/10.1007/s40840-022-01357-2

work page doi:10.1007/s40840-022-01357-2 2022
[18]

Zhu and C.-K

Zhu, Z.-H., Ren, C.-K.: On some determinant conjectures. arXiv:2409.07008 (2024). https://doi.org/10.48550/arXiv.2409.07008 17

work page doi:10.48550/arxiv.2409.07008 2024

[1] [1]

Borchardt, C.W.: Bestimmung der symmetrischen Verbindungen vermittelst ihrer erzeugenden Function. J. Reine Angew. Math. 53, 193–198 (1855)

work page

[2] [2]

Acta Arith

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

[3] [3]

Linear Multilinear Algebra 72(7), 1071–1077 (2024)

Guo, X., Li, X., Tao, Z., Wei, T.: The eigenvectors-eigenvalues identity and Sun’s conjectures on determinants and permanents. Linear Multilinear Algebra 72(7), 1071–1077 (2024). https://doi.org/10.1080/03081087.2023.2172380

work page doi:10.1080/03081087.2023.2172380 2024

[4] [4]

Graduate Texts in Mathematics, vol

Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990)

work page 1990

[5] [5]

Electron

Ishikawa, M., Kawamuko, H., Okada, S.: A Pfaffian-Hafnian analogue of Bor- chardt’s identity. Electron. J. Combin. 12, Research Paper N9 (2005)

work page 2005

[6] [6]

Krattenthaler, C.: Advanced determinant calculus. Sem. Lothar. Combin. 42, Article B42q (1999)

work page 1999

[7] [7]

Luo, X.-Q., Sun, Z.-W.: Legendre symbols related to certain determinants. Bull. Malays. Math. Sci. Soc. 46, Article 119 (2023). https://doi.org/10.1007/s40840- 023-01505-2 16

work page doi:10.1007/s40840- 2023

[8] [8]

Indian J

Luo, X.-Q., Xia, W.: Legendre symbols related toD p(b,1). Indian J. Pure Appl. Math. (2025). https://doi.org/10.1007/s13226-025-00882-7

work page doi:10.1007/s13226-025-00882-7 2025

[9] [9]

Encyclopedia of Mathematics and Its Applications, vol

Minc, H.: Permanents. Encyclopedia of Mathematics and Its Applications, vol. 6. Addison-Wesley, Reading (1978)

work page 1978

[10] [10]

Morley, F.: Note on the congruence 2 4n ≡(−1) n2n n (mod (2n+ 1) 3). Am. J. Math. 17(2), 168–170 (1895). https://doi.org/10.2307/2369517

work page doi:10.2307/2369517

[11] [11]

She, Y.-F., Sun, Z.-W.: Some determinants involving binary forms. Bull. Malays. Math. Sci. Soc. 48, Article 132 (2025). https://doi.org/10.1007/s40840-025-01900- x

work page doi:10.1007/s40840-025-01900- 2025

[12] [12]

arXiv:2208.12167 (2022)

She, Y.-F., Sun, Z.-W., Xia, W.: A novel permanent identity with applications. arXiv:2208.12167 (2022). https://doi.org/10.48550/arXiv.2208.12167

work page doi:10.48550/arxiv.2208.12167 2022

[13] [13]

Finite Fields Appl

Sun, Z.-W.: On some determinants with Legendre symbol entries. Finite Fields Appl. 56, 285–307 (2019). https://doi.org/10.1016/j.ffa.2018.12.004

work page doi:10.1016/j.ffa.2018.12.004 2019

[14] [14]

Sun, ‘On some determinants and permanents’,Acta Math

Sun, Z.-W.: On some determinants and permanents. Acta Math. Sinica Chin. Ser. 67(2), 286–295 (2024). https://doi.org/10.12386/A20220195

work page doi:10.12386/a20220195 2024

[15] [15]

Electron

Wang, H., Sun, Z.-W.: Proof of a conjecture involving derangements and roots of unity. Electron. J. Combin. 30(2), Article P2.1 (2023). https://doi.org/10.37236/11377

work page doi:10.37236/11377 2023

[16] [16]

Wang, H., Sun, Z.-W.: On certain determinants and related Legen- dre symbols. Bull. Malays. Math. Sci. Soc. 47, Article 58 (2024). https://doi.org/10.1007/s40840-024-01650-2

work page doi:10.1007/s40840-024-01650-2 2024

[17] [17]

Wu, H.-L., She, Y.-F., Ni, H.-X.: A conjecture of Zhi-Wei Sun on determi- nants over finite fields. Bull. Malays. Math. Sci. Soc. 45, 2405–2412 (2022). https://doi.org/10.1007/s40840-022-01357-2

work page doi:10.1007/s40840-022-01357-2 2022

[18] [18]

Zhu and C.-K

Zhu, Z.-H., Ren, C.-K.: On some determinant conjectures. arXiv:2409.07008 (2024). https://doi.org/10.48550/arXiv.2409.07008 17

work page doi:10.48550/arxiv.2409.07008 2024