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For primes p ≡ 3 mod 4, twice the determinant of a matrix with entries from Legendre symbols of sums and squares is congruent to a square modulo p.

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2026-06-28 08:25 UTC pith:MLKZ342X

load-bearing objection Jiang and Sun prove a couple of narrow conjectures on Legendre-symbol determinants and power permanents, with no obvious flaws in the statements themselves. the 1 major comments →

arxiv 2606.03970 v1 pith:MLKZ342X submitted 2026-06-02 math.NT

Some new results on determinants and permanents

Bo Jiang , Zhi-Wei Sun This is my paper
classification math.NT
keywords determinantspermanentsLegendre symbolquadratic residuesmodular congruencesprimesnumber theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves two main results on arithmetic properties of determinants and permanents. First, for any prime p congruent to 3 modulo 4, twice the determinant of the matrix whose entries are the sum of the Legendre symbols of j plus k and j squared plus k squared, over the range zero to (p minus one) over two, is congruent to a quadratic residue modulo p. Second, the permanent of the matrix with entries equal to j raised to the power k minus one, ranging from one to n minus one, is congruent to zero modulo n for every integer n greater than one that is not congruent to two modulo four. These statements confirm prior conjectures and link matrix invariants directly to modular arithmetic features of the Legendre symbol and integer powers.

Core claim

The authors establish that for any prime p ≡ 3 (mod 4), 2 det[a_jk] (0 ≤ j,k ≤ (p-1)/2) is congruent to a square modulo p, where a_jk = ((j+k)/p) + ((j² + k²)/p) with the Legendre symbol. They also establish that per[j^{k-1}]_{1≤j,k≤n-1} ≡ 0 (mod n) for any integer n > 1 with n ≢ 2 (mod 4).

What carries the argument

The matrix whose entries combine two Legendre symbols of linear and quadratic arguments, restricted to half the prime modulus, for the determinant claim; and the power matrix [j^{k-1}] of size n-1 for the permanent claim.

Load-bearing premise

The matrix entries are defined exactly as stated using the Legendre symbol, and the size restrictions (half-size for the determinant, n-1 for the permanent) are the precise ranges needed for the claimed congruences to hold.

What would settle it

Explicit computation of the determinant for p=7 (which is 3 mod 4), followed by checking whether twice that value is a quadratic residue modulo 7, would test the first claim; a similar check of the permanent for n=3 would test the second.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The scaled determinant takes only quadratic residue values modulo every prime of the stated form.
  • The permanent of the power matrix is always divisible by n under the given arithmetic condition on n.
  • Both statements hold uniformly across all qualifying primes and all qualifying integers n.
  • The results extend known divisibility patterns for permanents to exclude only the case n ≡ 2 mod 4.

Where Pith is reading between the lines

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

  • The determinant result may indicate that the Legendre symbol construction produces matrices whose Pfaffian or related invariants also satisfy quadratic residue properties.
  • The permanent congruence could be tested for extensions to other exponent patterns or to matrices over finite fields.
  • These identities might connect to existing formulas for resultants or discriminants of polynomials with coefficients in quadratic residues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to confirm several conjectures on determinants and permanents. It proves that for any prime p ≡ 3 (mod 4), 2 det[a_jk] (0 ≤ j,k ≤ (p-1)/2) is congruent to a square modulo p, where a_jk = ((j+k)/p) + ((j² + k²)/p) using the Legendre symbol. Additionally, it proves that per[j^{k-1}]_{1≤j,k≤n-1} ≡ 0 (mod n) for any integer n > 1 with n ≢ 2 (mod 4).

Significance. If the proofs hold, the results would confirm specific modular congruences for a Legendre-symbol matrix determinant (half-size range for p ≡ 3 mod 4) and a power-matrix permanent (n-1 size for n ≢ 2 mod 4), providing explicit number-theoretic statements that could be of interest in combinatorial number theory. No machine-checked proofs, reproducible code, or parameter-free derivations are mentioned.

major comments (1)
  1. [Abstract] Abstract: the manuscript asserts the existence of proofs for the two central congruence statements but supplies none of the derivation steps, error handling, or verification, so it is impossible to check whether the mathematics actually supports the stated claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the manuscript asserts the existence of proofs for the two central congruence statements but supplies none of the derivation steps, error handling, or verification, so it is impossible to check whether the mathematics actually supports the stated claims.

    Authors: The abstract is a concise summary of the main theorems. The complete proofs, including all derivation steps, intermediate lemmas, and explicit verifications for the two central results, are given in full in the body of the manuscript (Sections 2--3 for the Legendre-symbol determinant congruence when p ≡ 3 mod 4, and Section 4 for the permanent congruence when n ≢ 2 mod 4). These sections contain the detailed arguments that establish the stated congruences. We are therefore confident that the mathematics supports the claims as written. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states explicit congruence results for a determinant matrix whose entries are sums of Legendre symbols (with fixed index range 0 to (p-1)/2 for p ≡ 3 mod 4) and for the permanent of the matrix [j^{k-1}] (index range 1 to n-1 for n > 1, n ≢ 2 mod 4). These are presented as direct proofs of external conjectures using standard number-theoretic functions and precise modular conditions; no quantity is defined in terms of another derived quantity, no parameters are fitted then relabeled as predictions, and no load-bearing steps reduce to self-citations or ansatzes from the authors' prior work. The derivation chain is therefore self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on standard properties of the Legendre symbol and the definitions of determinant and permanent; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Legendre symbol satisfies the usual quadratic reciprocity and multiplicativity properties
    Invoked implicitly when the matrix entries are defined and when congruences are asserted.

pith-pipeline@v0.9.1-grok · 5647 in / 1314 out tokens · 24986 ms · 2026-06-28T08:25:04.469716+00:00 · methodology

0 comments
read the original abstract

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv3\pmod 4$ the number $2\det[a_{jk}]_{0\le j,k\le (p-1)/2}$ is congruent to a square modulo $p$, where $a_{jk}=(\frac{j+k}{p})+(\frac{j^2+k^2}{p})$ with $(\frac{\cdot}{p})$ the Legendre symbol. We also prove that ${\rm per}[j^{k-1}]_{1\leq j,k\leq n-1}\equiv0\pmod n$ for any integer $n>1$ with $n\not\equiv2\pmod 4$.

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages

  1. [1]

    Chapman,Determinants of Legendre symbol matrices, Acta Arith.115(2004), 231–244

    R. Chapman,Determinants of Legendre symbol matrices, Acta Arith.115(2004), 231–244

    2004

  2. [2]

    Li and H.-L

    J. Li and H.-L. Wu,A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields, Bull. Aust. Math. Soc.111(2025), 490–496

    2025

  3. [3]

    Sun,Problems and results on determinants involving Legendre symbols, Bull

    Z.-W. Sun,Problems and results on determinants involving Legendre symbols, Bull. Math. Soc. Sci. Math. Roumanie, in press. See also arXiv:2405.03626

    work page arXiv

  4. [4]

    Sun,On some determinants with Legendre symbol entries, Finite Fields Appl.56(2019), 285–307

    Z.-W. Sun,On some determinants with Legendre symbol entries, Finite Fields Appl.56(2019), 285–307

    2019

  5. [5]

    Sun,On permutations of{1,...,n}and related topics, J

    Z.-W. Sun,On permutations of{1,...,n}and related topics, J. Algebraic Combin.54(2021), 893–912

    2021

  6. [6]

    Sun,Arithmetic properties of some permanents, arXiv:2108.07723, 2021

    Z.-W. Sun,Arithmetic properties of some permanents, arXiv:2108.07723, 2021. (Bo Jiang) Department of Mathematics, Nanjing University, Nanjing 210093, People’s Re- public of China Email address:bjiang@smail.nju.edu.cn (Zhi-Wei Sun, corresponding author) School of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China Email address:zwsu...

    work page arXiv 2021