{
  "abstract": "We determine all positive integer solutions to $p^x+(2p+1)^y=z^2$, where $p$ is an odd prime and $|x-y|=1$. The unique solution is $(x,y,z)=(2,1,p+1)$ except when $p=7$, for which there is one additional solution, $(x,y,z)=(4,3,76)$. The proof reduces the nontrivial factorization case to an auxiliary equation, bounds its exponents using an AM--GM inequality, and isolates the exceptional solution with a divisibility argument modulo $2k^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18608",
  "author_total": 1,
  "authors": [
    {
      "name": "Subhasis Panda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkm1qd663b701.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18608",
  "fulltext_url": null,
  "id": "bkm1qd663b701",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18608",
  "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": "W7203859588",
  "pdf_url": "https://arxiv.org/pdf/2608.18608",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkm1qd663b701.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.18608",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18608"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Diophantine Equation $p^x+ (2p+1)^y =z^2$ with Consecutive Exponents",
  "title_text": "On the Diophantine Equation p^x+ (2p+1)^y =z^2 with Consecutive Exponents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/bkm1qd663b701",
  "venue": "arXiv",
  "versions": []
}