{
  "abstract": "Let $f_{0,1,4}(n)$ denote the number of partitions of $n$ into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let $f_{0,2,3}(n)$ be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime $p \\equiv 3 \\pmod 4$ there are explicit non-negative integers $a(p)$ and $b(p)$, determined by $20a(p) \\equiv -9$ and $20b(p) \\equiv -1 \\pmod{p^2}$, such that $f_{0,1,4}(p^2m + a(p))$ and $f_{0,2,3}(p^2m + b(p))$ are congruent modulo 3 to $f_{0,1,4}(m)$ and $f_{0,2,3}(m)$ when $p \\equiv \\pm 1 \\pmod 5$, and to $f_{0,2,3}(m)$ and $f_{0,1,4}(m)$ when $p \\equiv \\pm 2 \\pmod 5$. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo $p$, and interchanged otherwise. The smallest cases are $f_{0,1,4}(9m) \\equiv f_{0,2,3}(m)$ and $f_{0,2,3}(9m+4) \\equiv f_{0,1,4}(m)$. The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of $(1-x)^{\\ell} \\equiv 1-x^{\\ell} \\pmod{\\ell}$ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that $-1$ is a quadratic non-residue modulo $p$. As corollaries we obtain, for each such $p$, a congruence with exceptions on a progression of modulus $p$, and an infinite family of self-similarity congruences on progressions of modulus $p^{dn}$ with $d = 2$ or $d = 4$ and constants $9(p^{dn}-1)/20$ and $(p^{dn}-1)/20$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22340",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiyuan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4m5gzn2w3zmjv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22340",
  "fulltext_url": null,
  "id": "4m5gzn2w3zmjv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22340",
  "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|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214041393",
  "pdf_url": "https://arxiv.org/pdf/2609.22340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4m5gzn2w3zmjv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A quadratic-residue dichotomy for two partition functions modulo 3",
  "title_text": "A quadratic-residue dichotomy for two partition functions modulo 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/4m5gzn2w3zmjv",
  "venue": "arXiv",
  "versions": []
}