{
  "abstract": "In this paper, we study two questions concerning perfect powers in $S$-unit equations and sparse representations of perfect powers. Corvaja-Zannier \\cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in $\\N$ having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation \\begin{equation*} y^d=1+c_1g^{m_1}+c_2g^{m_2}+c_3g^{m_3}, \\quad 0<m_1<m_2<m_3, \\end{equation*} where $c_i$ are $S$-units for a suitable set $S$. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at $S$-unit points with Roth's theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \\cite{bennettbugead2013perfect} by proving that, for every fixed integer base $g\\geq 2$, there exists a constant $q_0(g)$ such that, for every prime $q>q_0(g)$, no perfect $q$-th power admits a representation with five non-zero digits in base $g$. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08310",
  "author_total": 3,
  "authors": [
    {
      "name": "Darsana N"
    },
    {
      "name": "Parvathi S Nair"
    },
    {
      "name": "Sudhansu Sekhar Rout"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cj29gfdb288m6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08310",
  "fulltext_url": null,
  "id": "cj29gfdb288m6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08310",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221027373",
  "pdf_url": "https://arxiv.org/pdf/2610.08310",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cj29gfdb288m6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08310",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08310"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Perfect powers with few digits and $S$-unit coefficients",
  "title_text": "Perfect powers with few digits and S-unit coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/cj29gfdb288m6",
  "venue": "arXiv",
  "versions": []
}