{
  "abstract": "The present study explores integer and rational matrix solutions for the Diophantine equation $X^{4}+Y^{4}=2Z^{4}$, establishing that infinitely many solutions exist over different matrix rings, including $M_2(\\mathbb{Z})$, $M_3(\\mathbb{Z})$, $M_4(\\mathbb{Z}).$ Furthermore, expanding the scope beyond the integer domain, the analysis demonstrates that an infinite number of matrix solutions can be found in degree-4 and degree-16 field extensions, specifically within $M_4(\\mathbb{Q}(\\sqrt[4]{l}))$ and $M_4(\\mathbb{Q}(\\sqrt[4]{l_1}, \\sqrt[4]{l_2}))$ when the rational numbers involved lack perfect fourth powers and are not perfect squares.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32374",
  "author_total": 2,
  "authors": [
    {
      "name": "Swadhin Moharana"
    },
    {
      "name": "Pabitra Kumar Jena"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ex8pcjntygd8j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32374",
  "fulltext_url": null,
  "id": "ex8pcjntygd8j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32374",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214747568",
  "pdf_url": "https://arxiv.org/pdf/2609.32374",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ex8pcjntygd8j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32374",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32374"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Matrix Solutions to the Diophantine equation $X^4+ Y^4=2Z^4$",
  "title_text": "Matrix Solutions to the Diophantine equation X^4+ Y^4=2Z^4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ex8pcjntygd8j",
  "venue": "arXiv",
  "versions": []
}