{
  "abstract": "Let $p>5$ be an odd prime, put $n=(p-1)/2$, and let $\\left(\\frac{\\cdot}{p}\\right)$ be the Legendre symbol. Define $D_p^{\\pm}$ to be the following determinants \\[ \\begin{aligned} D_p^{\\pm}&:=\\det\\left[(i\\pm j)\\left(\\frac{i\\pm j}{p}\\right) \\right]_{1\\leq i,j\\leq n}. \\end{aligned} \\] In this article, we give explicit formulas for these determinants modulo $p$ in terms of Bernoulli numbers. As a consequence of Reinhart's recent counterexample to the Ankeny--Artin--Chowla conjecture, we find a prime $p$ such that $D_p^{\\pm} \\equiv 0 \\pmod p$. This gives a negative answer to a question of the Sun.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23684",
  "author_total": 1,
  "authors": [
    {
      "name": "Hang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7k9fyc208watb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23684",
  "fulltext_url": null,
  "id": "7k9fyc208watb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23684",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214010310",
  "pdf_url": "https://arxiv.org/pdf/2609.23684",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7k9fyc208watb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23684",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23684"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Two Legendre-symbol determinants of Sun and Bernoulli numbers",
  "title_text": "Two Legendre-symbol determinants of Sun and Bernoulli numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7k9fyc208watb",
  "venue": "arXiv",
  "versions": []
}