{
  "abstract": "Let $q$ be an odd prime power and put \\[ θ_q=\\frac{φ(q-1)}{q-1}. \\] An element of $\\Fq$ is called non-square non-primitive, or \\emph{NSNP}, if it is both a non-square and a non-primitive element. Let $\\ell$ be an odd prime divisor of $q-1$. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square $\\ell$th powers. Combining this estimate with a finite computation, we prove that $θ_q<4/15$ guarantees three consecutive NSNP elements. On the boundary $θ_q=4/15$, the only exceptions are $q\\in\\{31,61,121\\}$. As a further application, we show that for $\\operatorname{char}\\Fq>3$, the inequality $θ_q<8/35$ guarantees four consecutive NSNP elements, with $q=211$ as the unique exception on the boundary $θ_q=8/35$. Thus both constants are best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17267",
  "author_total": 2,
  "authors": [
    {
      "name": "Juncheng Zhou"
    },
    {
      "name": "Hongfeng Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9hj6n4qg47jv1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17267",
  "fulltext_url": null,
  "id": "9hj6n4qg47jv1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17267",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7169855610",
  "pdf_url": "https://arxiv.org/pdf/2607.17267",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9hj6n4qg47jv1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.17267",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17267"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Consecutive non-square non-primitive tuples in finite fields",
  "title_text": "Consecutive non-square non-primitive tuples in finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/9hj6n4qg47jv1",
  "venue": "arXiv",
  "versions": []
}