{
  "abstract": "We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\\equiv4,7,8\\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\\equiv8\\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $λ$-division before taking the trace, where $λ=1-ω$ and $ω$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14893",
  "author_total": 2,
  "authors": [
    {
      "name": "Ashay Burungale"
    },
    {
      "name": "Ye Tian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/65ksrk5aqc6rn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14893",
  "fulltext_url": null,
  "id": "65ksrk5aqc6rn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14893",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7213300894",
  "pdf_url": "https://arxiv.org/pdf/2609.14893",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/65ksrk5aqc6rn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.14893",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14893"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A proof of Sylvester's conjecture",
  "title_text": "A proof of Sylvester's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/65ksrk5aqc6rn",
  "venue": "arXiv",
  "versions": []
}