{
  "abstract": "We investigate the properties of matrices obtained from a geometric transformation of the first N rows of Pascal's triangle. For N > 2, their congruence properties form a Pascal tiling, that is, a perfect alternation between entries congruent to 0 mod N and the others, if and only if N is prime.\n\nThis result yields an alternative proof of the classical congruence LN -1 ≡ 0 mod N for prime N , where LN denotes the N th Lucas number. Within the framework of the Pascal tiling theorem, this congruence can be expressed as a sum of entries lying along a diagonal of one of the matrices under consideration; when N is prime, each of these entries is congruent to 0 mod N . By contrast, for Fibonacci pseudoprimes, the sum remains congruent to 0 mod N while at least one of its terms is not.\n\nFinally, these results are interpreted in terms of decompositions of binomial coefficients and extended to multinomial coefficients, leading to a study of the associated symmetries. This perspective highlights the case where N is a prime power and clarifies the conditions under which a Pascal tiling arises.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08343",
  "author_total": 2,
  "authors": [
    {
      "name": "Etienne Rousseau"
    },
    {
      "name": "Guillaume Rousseau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e51wecnqwv1sq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08343",
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  "id": "e51wecnqwv1sq",
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  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
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  "pdf_url": "https://hal.science/hal-05735606/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e51wecnqwv1sq.ris",
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    {
      "first_seen": "2026-10-30T12:17:27Z",
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      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08343"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
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    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pascal tiling and congruences modulo N in Pascal's triangle",
  "title_text": "Pascal tiling and congruences modulo N in Pascal's triangle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/e51wecnqwv1sq",
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}