The open half of Sun's Conjecture 4.10(ii): nonvanishing of det[((j+k)/(p))+(tfracj²+k²p)] for every prime p ≡ 3 (mod 4) below 1500
Abstract
Let p be an odd prime, n=(p-1)/2, and let A(p)=[leg(j+k, p)+legj²+k²p]_(0 ≤ j,k ≤ n) be the integer matrix of Legendre symbols studied by Jiang and Sun. Sun's Conjecture 4.10(ii) asserts that 2det A(p) is a quadratic residue modulo p whenever p ≡ 3 (mod 4). Jiang and Sun proved ((2det A(p))/(p)) ≠ -1 and remarked that they are unable to prove p ∤ det A(p); as a quadratic residue is by definition nonzero, that nonvanishing is the entire remaining content of the conjecture. We prove p ∤ det A(p) for every prime p ≡ 3 (mod 4) with p ≤ 1500 — 122 primes, matrices of side up to 750 — by exhibiting for each such p a right inverse of A(p) over 𝔽ₚ and verifying the product; the verification is machine-checked, so the window is a theorem rather than the report of a computation. With Jiang and Sun's theorem this settles Conjecture 4.10(ii) for those 122 primes. We also show that the hypothesis p ≡ 3 (mod 4) cannot be dropped and that Jiang and Sun's theorem does not by itself imply the conjecture: 5 | det A(5) and 13 | det A(13), each witnessed by an explicit kernel vector, while at p=5 the conclusion ((2det A(5))/(5)) ≠ -1 still holds, the symbol being 0. Outside the formal development, an independent sweep confirms the full conjecture for all 399 primes p ≡ 3 (mod 4) below 6000, and records the behaviour of the same matrix at p ≡ 1 (mod 4). Finally we point out that a Vandermonde factor in Jiang and Sun's own displayed identity reduces p ∤ det A(p) to p ∤ det C₂ on a matrix of side (p+1)/4.
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Claim ledger
Stated results
SD1candidate2026-09-03
Sun's Conjecture 4.10(ii): p does not divide det[((j+k)/p)+((j²+k²)/p)]_(0<=j,k<=(p-1)/2) for EVERY prime p = 3 (mod 4) with p <= 1500 (122 primes, matrices up to 750x750) – the half Jiang-Sun could not prove, kernel-checked
SD2routine2026-09-03
Certificate soundness: a right inverse over ZMod p, checked entrywise in flat Nat arithmetic, forces p to not divide the determinant of the integer Legendre-symbol matrix – with Euler's criterion identifying the Nat table with the reduction of legendreSym
SD3routine2026-09-03
The checker is not vacuous: sunCert returns false at p = 5 and p = 13 (where the matrix really is singular mod p) and true at p = 7
SD4routine2026-09-03
Too-large control: drop the hypothesis p = 3 (mod 4) and nonvanishing is false – 5 divides det A(5) and 13 divides det A(13), each witnessed by an explicit kernel vector; and at p = 5 Jiang-Sun's proved conclusion (2 det A|p)!= -1 STILL HOLDS (the symbol is 0), so Theorem 1.1 of arXiv:2606.03970v1 does not imply Conjecture 4.10(ii)
SD5candidate2026-09-03
C sweep: Sun's Conjecture 4.10(ii) holds IN FULL – 2 det A is a nonzero quadratic residue mod p – for all 399 primes p = 3 (mod 4) with 3 <= p <= 5987; no zero determinant and no non-residue occurs
This ledger entry is reported in prose and is not bound to a Lean theorem.SD6candidate2026-09-03
The congruence hypothesis is doing real work: for the SAME matrix at p = 1 (mod 4), p divides det A at exactly p = 5 and p = 13 among the 211 primes 5 <= p <= 2969, and (2 det A|p) = -1 for 127 of those 211 primes
This ledger entry is reported in prose and is not bound to a Lean theorem.SD7known2026-09-03
Independent numerical confirmation of Sun's Conjecture 4.10(i) (p = 1 mod 4, all four sign choices): 844 determinants for the 211 primes 5 <= p <= 2969, every one with 2 det a nonzero quadratic residue
This ledger entry is reported in prose and is not bound to a Lean theorem.SD8known2026-09-03
Compute-first gate: Chapman's 2004 evaluation, Sun's 2019 Sₚ/Tₚ relations, and Jiang-Sun's own reduction det A = (-1)^((p+1)/4) det(H)² det(C₂)² (mod p) all reproduced from their definitions with independent code before any claim was made
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Korea Superintelligence Labs
- Source context
- Let p be an odd prime, (a|p) the Legendre symbol, and set n = (p-1)/2. Sun's Conjecture 4.10 (Zhi-Wei Sun, *Problems and results on determinants involving Legendre symbols*, arXiv:2405.03626v8, last conjecture of section 4) has two parts. Part (ii) is this family's object:
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