{
  "abstract": "We determine the algebra generated by the normalized parity-twisted crank moments and describe it explicitly as a proper subalgebra of the quasi-modular forms on $Γ_0(2)$ closed under $D=q\\frac{d}{dq}$. We derive congruences from differential identities and construct a prime-detecting expression involving only the normalized twisted second moment, its products, and its derivatives. Its coefficient of $q^n$ vanishes if and only if $n$ is prime, for every $n\\ge2$. We prove that its highest weight, eight, is minimal among prime-detecting elements of this algebra. Finally, we identify the same algebra as the one generated by MacMahon's functions $B_k$, obtaining a corresponding prime-detecting expression in $B_1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37301",
  "author_total": 1,
  "authors": [
    {
      "name": "Soon-Yi Kang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4n0qm8m9zh23f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37301",
  "fulltext_url": null,
  "id": "4n0qm8m9zh23f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37301",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.37301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4n0qm8m9zh23f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression",
  "title_text": "The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/4n0qm8m9zh23f",
  "venue": "arXiv",
  "versions": []
}