{
  "abstract": "Lucas' theorem states that for a prime $p$, integers $a,b\\geq0$ and $0\\leq r,s\\leq p-1$, one has $\\binom{pa+r}{pb+s}\\equiv\\binom{a}{b}\\binom{r}{s}\\bmod p$. Let $D(p^n)$ be the set of pairs $(r,s)$ with $s\\leq r$ for which this congruence holds modulo $p^n$ for all $a,b\\geq0$. Rowland characterized $D(p^2)$ in terms of harmonic sums and symmetries of Pascal's triangle. For $n<p$, we give a criterion for a point $(r,s)$ to lie in $D(p^n)$ in terms of generalized harmonic sums of order at most $n-1$. We show that $D(p^n)$ is invariant under the reflection $(r,s)\\mapsto(r,r-s)$ and the rotation $(r,s)\\mapsto(p-1-s,r-s)$, so that with every point it contains its full orbit under the dihedral group $D_3$. Applied to the vertices, our criterion gives a result of Jacobsthal-Ljunggren type modulo $p^n$. In particular, we recover Zhao's result $(0,0)\\in D(p^4)$ if and only if $p$ is a Wolstenholme prime. We also study the centroid and the edge midpoints, whose membership in $D(p^3)$ and $D(p^4)$ is governed by Bernoulli numbers and polynomials, Wolstenholme primes and super Wieferich primes. In particular, for $p=205129$ the centroid lies in $D(p^3)$, answering a question of Rowland.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25248",
  "author_total": 1,
  "authors": [
    {
      "name": "Suneel Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5rkjaj0pcmkjs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25248",
  "fulltext_url": null,
  "id": "5rkjaj0pcmkjs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25248",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223009536",
  "pdf_url": "https://arxiv.org/pdf/2610.25248",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5rkjaj0pcmkjs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25248",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25248"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lucas' Theorem modulo $p^n$ via Harmonic sums",
  "title_text": "Lucas' Theorem modulo p^n via Harmonic sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/5rkjaj0pcmkjs",
  "venue": "arXiv",
  "versions": []
}