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A Determinant Congruence Conjectured by Sun

Number Theory 2026-05-29 v3

Abstract

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let n>3n>3 and let c,dZc,d\in\Z. If nn is composite, then det[(i2+cij+dj2)n2]0i,jn10(modn2) \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2} with no condition on cc and dd. If n=pn=p is prime, the same congruence holds whenever the Legendre symbol \legdp\leg{d}{p} is 1-1. For composite nn, a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime n=pn=p, we estimate the rank of the matrix modulo pp. The required rank defect follows from a coefficient cancellation obtained from the involution td/tt\mapsto d/t on \Fp×\Fp^\times and the condition \legdp=1\leg{d}{p}=-1.

Keywords

Cite

@article{arxiv.2605.19486,
  title  = {A Determinant Congruence Conjectured by Sun},
  author = {Yutong Zhang and Yaoran Yang},
  journal= {arXiv preprint arXiv:2605.19486},
  year   = {2026}
}

Comments

Accepted for publication in the Bulletin of the Australian Mathematical Society