{
  "abstract": "We classify, for every prime power $q$ and every $e\\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\\mathbb F_{q^e}$. Writing $\\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\\gcd(r,q-1)=1$, $(-a)^{\\ell_e(q)}\\ne1$, and $r\\ell_h(q)\\equiv1\\pmod{\\ell_e(q)}$ for some $1\\leqslant h<e$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\\gcd(d,\\ell_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20354",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiang Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fv4sxp7b3njbx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20354",
  "fulltext_url": null,
  "id": "fv4sxp7b3njbx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20354",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7213602794",
  "pdf_url": "https://arxiv.org/pdf/2609.20354",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fv4sxp7b3njbx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20354",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20354"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields",
  "title_text": "A complete classification of permutation binomials of the form X^r(X^q−1+a) over finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/fv4sxp7b3njbx",
  "venue": "arXiv",
  "versions": []
}