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 and let . If is composite, then with no condition on and . If is prime, the same congruence holds whenever the Legendre symbol is . For composite , a polynomial determinant is divisible by two Vandermonde factors; after specialisation, their product already yields the required square divisor. For prime , we estimate the rank of the matrix modulo . The required rank defect follows from a coefficient cancellation obtained from the involution on and the condition .
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