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REVIEW 2 major objections 2 minor 9 references

D_a(0) vanishes exactly when the prime p is 3 mod 4 and χ(a n!)=1.

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2026-07-03 07:19 UTC pith:VYKNFWPA

load-bearing objection The paper proves Sun's conjecture via explicit Pfaffian factorizations and generalizes the vanishing and square results to arbitrary a, with the central claims internally consistent though the auxiliary skew-symmetric construction needs verification. the 2 major comments →

arxiv 2607.01695 v1 pith:VYKNFWPA submitted 2026-07-02 math.NT

A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei

Hong-Ge Chen , Fei Liu This is my paper
classification math.NT
keywords determinantsLegendre symbolPfaffianSun conjecturequadratic characterodd primesmatrix factorization
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 studies two families of determinants whose entries involve the Legendre symbol χ applied to quadratic expressions in i and j scaled by a parameter a. It establishes an if-and-only-if criterion for when the constant term D_a(0) is zero. For primes congruent to 3 modulo 4 the authors construct explicit Pfaffian factorizations that reduce both determinants to squares or to squares times linear polynomials in x. The sign factor s_p and the value of χ(a n!) determine whether the scaled determinant is a square or factors through a linear term. This confirms Sun's conjecture in the special case a equal to n factorial.

Core claim

We prove D_a(0)=0 ⇔ p≡3 (mod 4) and χ(a n!)=1. For p≡3 mod4 we also give explicit Pfaffian-square factorizations of D_a(x) and D_a^{(0)}(x). If χ(a n!)=1, then s_p D_a(x)/x = s_p D_a^{(0)}(x) is a positive integer square. If χ(a n!)=-1, then s_p D_a(x)=σ²(nx-1), s_p D_a^{(0)}(x)=-σ²(n+(2n+1)x). The case a=n! settles Sun's Conjecture 4.1.

What carries the argument

Pfaffian-square factorizations of the matrices with entries x + χ(i² - a j) when p ≡ 3 mod 4.

Load-bearing premise

The matrices with entries x + χ(i² - a j) possess the skew-symmetry or other algebraic properties required for Pfaffian factorizations to hold when p ≡ 3 mod 4.

What would settle it

Compute the 3 by 3 determinant D_1(0) for the prime p=7 (n=3) and verify it is nonzero, since χ(1·6) = χ(6) = -1 for p=7; if it vanishes the claim fails.

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

If this is right

  • D_a(0) equals zero precisely when p ≡ 3 mod 4 and χ(a n!)=1.
  • Explicit Pfaffian factorizations exist for D_a(x) and D_a^{(0)}(x) whenever p ≡ 3 mod 4.
  • When χ(a n!)=1 the scaled determinants s_p D_a(x)/x and s_p D_a^{(0)}(x) are identical positive integer squares.
  • When χ(a n!)=-1 the scaled determinants equal σ²(nx-1) and -σ²(n+(2n+1)x) for a positive integer σ.

Where Pith is reading between the lines

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

  • The Pfaffian approach may extend to evaluating other determinants built from quadratic characters.
  • The generalization covers all a in the multiplicative group of the prime field, not just the conjectured case a = n!.
  • Explicit formulas for the integer σ could be derived from the Pfaffian entries for small primes.
  • Similar determinant identities might appear in the study of character sums over finite fields.

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

2 major / 2 minor

Summary. The paper defines D_a(x) = det_{1≤i,j≤n} (x + χ(i² - a j)) and D_a^{(0)}(x) = det_{0≤i,j≤n} (x + χ(i² - a j)), where p is an odd prime, n=(p-1)/2, and χ is the Legendre symbol mod p with χ(0)=0. It proves D_a(0)=0 ⇔ p≡3 (mod 4) and χ(a n!)=1. For p≡3 (mod 4) it supplies explicit Pfaffian-square factorizations of both determinants. With s_p = (-1)^{⌊(p+1)/8⌋}, the paper states that if χ(a n!)=1 then s_p D_a(x)/x = s_p D_a^{(0)}(x) is a positive integer square, while if χ(a n!)=-1 then s_p D_a(x) = σ²(n x -1) and s_p D_a^{(0)}(x) = -σ²(n + (2n+1)x) for a positive integer σ. The case a=n! settles Sun's Conjecture 4.1.

Significance. If the explicit Pfaffian constructions are correct, the work supplies a linear-algebraic proof of Sun's conjecture together with a uniform generalization to all a ∈ F_p^×, including closed-form expressions that distinguish the two values of χ(a n!). The appearance of Pfaffians in this arithmetic setting is novel and, if the auxiliary skew-symmetric matrices are exhibited and verified, could open further applications to other character-sum determinants.

major comments (2)
  1. [Abstract (statements on Pfaffian-square factorizations) and the section containing the explicit constructions] The matrix M with entries M_{ij} = x + χ(i² - a j) satisfies M_{ji} + M_{ij} = 2x + χ(j² - a i) + χ(i² - a j) which is not identically zero, so M is not skew-symmetric and det(M) is not automatically a Pfaffian square. The claimed explicit Pfaffian-square factorizations for p≡3 (mod 4) therefore require an auxiliary construction (larger block matrix or signed similarity) whose correctness must be verified entrywise or by direct expansion in the cases χ(a n!)=±1; this verification is load-bearing for all subsequent statements about squares and the settlement of the conjecture.
  2. [Theorem stating D_a(0)=0 ⇔ …] The equivalence D_a(0)=0 ⇔ p≡3 (mod 4) and χ(a n!)=1 is asserted as proved from the matrix definitions. Because the 'if' direction appears to rely on the Pfaffian factorization (whose auxiliary matrix is not yet shown to be skew-symmetric), the logical dependence between the vanishing criterion and the factorization must be made explicit; otherwise the equivalence rests on an unverified step.
minor comments (2)
  1. The definition of s_p should appear in the introduction rather than only in the abstract.
  2. The index ranges (1 to n versus 0 to n) should be restated when D_a^{(0)}(x) is first used after the definition of D_a(x).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the logical structure and verification of the Pfaffian constructions can be made more explicit. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract (statements on Pfaffian-square factorizations) and the section containing the explicit constructions] The matrix M with entries M_{ij} = x + χ(i² - a j) satisfies M_{ji} + M_{ij} = 2x + χ(j² - a i) + χ(i² - a j) which is not identically zero, so M is not skew-symmetric and det(M) is not automatically a Pfaffian square. The claimed explicit Pfaffian-square factorizations for p≡3 (mod 4) therefore require an auxiliary construction (larger block matrix or signed similarity) whose correctness must be verified entrywise or by direct expansion in the cases χ(a n!)=±1; this verification is load-bearing for all subsequent statements about squares and the settlement of the conjecture.

    Authors: The manuscript supplies explicit auxiliary skew-symmetric matrices (constructed via block augmentation and signed permutation similarity in the Pfaffian section) together with the verification that their Pfaffians squared recover D_a(x) and D_a^{(0)}(x). The verification uses the multiplicative property of χ and case-by-case expansion on the value of χ(a n!). We will revise by adding a dedicated verification subsection that records the entrywise identities and the direct expansion for both signs of χ(a n!), rendering the auxiliary construction fully transparent. revision: yes

  2. Referee: [Theorem stating D_a(0)=0 ⇔ …] The equivalence D_a(0)=0 ⇔ p≡3 (mod 4) and χ(a n!)=1 is asserted as proved from the matrix definitions. Because the 'if' direction appears to rely on the Pfaffian factorization (whose auxiliary matrix is not yet shown to be skew-symmetric), the logical dependence between the vanishing criterion and the factorization must be made explicit; otherwise the equivalence rests on an unverified step.

    Authors: The proof of the equivalence is organized as follows. The implication 'D_a(0)=0 ⇒ p≡3 (mod 4) and χ(a n!)=1' is obtained from the matrix definition by evaluating the determinant at x=0 and applying character-sum identities, without any reference to Pfaffians. The converse relies on the factorization: when p≡3 (mod 4) and χ(a n!)=1 the factorization yields D_a(x)=x·(square), hence D_a(0)=0; when χ(a n!)=-1 the factorization yields D_a(x)=(n x-1)·(square), hence D_a(0)≠0. We will revise the theorem statement and proof to display this logical order explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the matrices D_a(x) and D_a^{(0)}(x) entrywise via the Legendre symbol and proves equivalences and factorizations as direct theorems. No parameters are fitted to data and then renamed as predictions; no self-citations are load-bearing for the central claims; no ansatz is smuggled via prior work; and no result is renamed or defined in terms of itself. The Pfaffian-square factorizations are asserted as explicit constructions for the p≡3 mod 4 case, independent of the target statements about D_a(0) or the conjecture settlement. The derivation chain therefore does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies entirely on standard mathematical axioms from number theory and linear algebra; no free parameters or new entities introduced.

axioms (2)
  • standard math Standard properties of the Legendre symbol χ over finite fields F_p
    Used to define the entries of the matrices D_a(x) and D_a^{(0)}(x).
  • standard math Existence and algebraic properties of Pfaffians for matrices with appropriate symmetry
    Invoked to obtain the square factorizations when p ≡ 3 mod 4.

pith-pipeline@v0.9.1-grok · 5824 in / 1568 out tokens · 36778 ms · 2026-07-03T07:19:41.693685+00:00 · methodology

0 comments
read the original abstract

Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\chi=(\frac{\cdot}{p})$, with $\chi(0)=0$. For $a\in\mathbb F_p^\times$ define \[ D_a(x)=\det_{1\le i,j\le n}(x+\chi(i^2-aj)), \qquad D_a^{(0)}(x)=\det_{0\le i,j\le n}(x+\chi(i^2-aj)). \] We prove \[ D_a(0)=0 \quad\Longleftrightarrow\quad p\equiv 3 \pmod 4 \quad\text{and}\quad \chi(a n!)=1. \] For $p\equiv3\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\lfloor(p+1)/8\rfloor}$. If $\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. If $\chi(a n!)=-1$, then there is a positive integer $\sigma$ such that \[ s_pD_a(x)=\sigma^2(nx-1),\qquad s_pD_a^{(0)}(x)=-\sigma^2\bigl(n+(2n+1)x\bigr). \] The case $a=n!$ settles Sun's Conjecture 4.1.

discussion (0)

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 2 internal anchors

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