{
  "abstract": "Let $p$ be a prime number and let $n\\geq 1$ be an integer. We completely solve the exponential Diophantine equation $p^{x}+6^{y}=z^{2n}$ in positive integers $x,y,z$ for every prime $p$ such that the Legendre symbol $\\leg{6}{p}$ equals $-1$, that is, for $p\\equiv 7,11,13,17\\pmod{24}$: the equation has no solution, with the single exception $(p,n,x,y,z)=(13,1,1,2,7)$. Since these residue classes contain half of all primes in the sense of Dirichlet density, this settles the equation for one prime out of two, uniformly in the exponent $n$. Two ingredients of the proof have independent interest. First, we show that the Ramanujan--Nagell type equation $p^{x}=2\\cdot 6^{m}+1$ has no solution with $x\\geq 2$, using Zsigmondy's theorem on primitive prime divisors; this removes the extra congruence hypotheses that appeared in earlier work of the author on the case $n=1$ and solves the open problems formulated there. Secondly, we prove that $13^{x}+6^{y}=z^{2}$ has $(x,y,z)=(1,2,7)$ as its unique solution, thereby completely resolving the exceptional prime. The Legendre condition is sharp: we exhibit solutions for primes with $\\leg{6}{p}=+1$ realising each branch of the factorisation argument. All results are corroborated by extensive computational verification, and we state several open problems, including the complete classification of the solutions when $\\leg{6}{p}=+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09214",
  "author_total": 1,
  "authors": [
    {
      "name": "Pagdame Tiebekabe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajhw1qxk9cw69.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09214",
  "fulltext_url": null,
  "id": "ajhw1qxk9cw69",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09214",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212189462",
  "pdf_url": "https://arxiv.org/pdf/2609.09214",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajhw1qxk9cw69.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09214",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09214"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Diophantine equation $p^{x}+6^{y}=z^{2n}$: a complete solution for half of the primes",
  "title_text": "The Diophantine equation p^x+6^y=z^2n: a complete solution for half of the primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/ajhw1qxk9cw69",
  "venue": "arXiv",
  "versions": []
}