Sun's conjecture on det Cₚ(x): the unspecified coefficient dₚ determined, and both halves verified for every prime below 300

Abstract

Let p>3 be a prime, n=(p-1)/2, and let Cₚ(x)=[x+cᵢⱼ]_(1 ≤ i,j ≤ n) be the shifted Legendre-symbol matrix with c₁ⱼ=leg(j, p) and cᵢⱼ=leg(i-j, p) for i ≥ 2. Ren and Sun, having evaluated the companion matrix with leg(i+j, p) in place of leg(i-j, p), record a conjecture of the second author on det Cₚ(x) and state that their method does not reach it. For p ≡ 1 (mod 4) that conjecture reads det Cₚ(x)=leg(2, p)(aₚ'-(p+1)/(2)bₚ')+dₚx "for some integer dₚ", where varepsilonₚ^((2-leg(2, p))h(p))=aₚ'+bₚ'sqrt p; no value of dₚ is proposed, there and, as far as we could find, anywhere else. We propose one: dₚ=(p+1)/(2)aₚ'-p bₚ'. It carries no factor leg(2, p), unlike the constant term, and the variant that does carry one is false already at p=13. We verify the completed conjecture exactly for all 80 primes p ≡ 1 (mod 4) with p ≤ 1000, and for the 29 such primes p ≤ 293 we prove — for all x ∈ ℤ at once — a restatement of it that mentions no class number and no fundamental unit, from which the conjecture with dₚ named follows at those primes granting Vsemirnov's evaluation of Chapman's determinant. For p ≡ 3 (mod 4) we prove the corresponding class-number-free statement for the 31 primes 7 ≤ p ≤ 283, and Sun's conjecture follows there granting Wang and Wu's evaluation of the all-ones-row variant; together the two windows cover every prime 5 ≤ p ≤ 300. Those two rewritings are what make the statements elementary enough to be machine-checked; the checking is done in Lean 4 against Mathlib. Outside the formal development we record what we believe to be the first verification range for this conjecture: all 166 primes 5 ≤ p ≤ 1000, no exceptions. We also observe that det Cₚ(0) is a single cofactor of Chapman's matrix, hence a product of two Pfaffians when p ≡ 3 (mod 4), and offer that as a route to a proof.

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