{
  "abstract": "Author: Sayan Kumar Mitra, Class 11, Rajendra Vidyalaya, Jamshedpur, Jharkhand, India.\n\nIndependent Researcher in Number Theory.\n\nTITLE: Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) modulo 7\n\nABSTRACT:\n\nLet p_{4,1}(n) denote the number of partitions of n where parts ≡ ±1 mod 4 are distinct and counted with specific weights.\n\nIn this paper, I proved that there are infinitely many n such that p_{4,1}(n) ≡ 0 mod 7.\n\nThe proof uses the theory of modular forms modulo 7. I show that the generating function G(q) for p_{4,1}(n) is congruent modulo 7 to a product H(q) = ∏_{m≥1} 1/(1 - q^{m}) with modifications, which exhibits modular properties.\n\nThis is related to Ramanujan's famous partition congruences p(5n+4) ≡ 0 mod 5.\n\nConjecture: There exists an infinite family of congruences of the form p_{4,1}(l^k n + δ_k) ≡ 0 mod 7 for primes l.\n\nKeywords: p41 partitions, partitions mod 7, Ramanujan congruences, Sayan Kumar Mitra, Rajendra Vidyalaya Jamshedpur, Number Theory\n\nHigh School Research Paper",
  "abstract_shown": true,
  "affiliations": [
    "Rajendra Vidyalaya, Jamshedpur, Jharkhand, India - Class 11 Student (Independent Researcher)"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Sayan Kumar Mitra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19q3tdrbe91te.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23465776",
  "fulltext_url": null,
  "id": "19q3tdrbe91te",
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  "landing_url": "https://zenodo.org/records/23465776",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
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  "pdf_url": "https://zenodo.org/api/records/23465776/files/Sayan_p%7B4,1%7D%20mod%207.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19q3tdrbe91te.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:17Z",
      "native_id": "23465776",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23465776"
    }
  ],
  "subjects": [],
  "title": "Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur",
  "title_text": "Infinitely Many Ramanujan-type Congruences for p_{4,1}(n) mod 7 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur",
  "type": "preprint",
  "updated_at": "2026-11-02T12:21:17Z",
  "url": "https://research.draconitic.com/paper/19q3tdrbe91te",
  "venue": "Zenodo",
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}