Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 38 min ago
PreprintCC BY 4.0arXiv

Matrix Solutions to the Diophantine equation X4+ Y4=2Z4

  • Swadhin Moharana
  • Pabitra Kumar Jena
Open at sourcePDFDOICite

Abstract

The present study explores integer and rational matrix solutions for the Diophantine equation X4+Y4=2Z4, establishing that infinitely many solutions exist over different matrix rings, including M2(ℤ), M3(ℤ), M4(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 M4(ℚ(√[4]l)) and M4(ℚ(√[4]l1, √[4]l2)) when the rational numbers involved lack perfect fourth powers and are not perfect squares.

Related

  1. Elliptic solutions to matrix CKP equation
    A. Zabrodin
    arXivPreprintarXiv licenceFull textMathematical physicsStatistical & nonlinear physicsarXiv
  2. Some Methods to Solve the Diophantine Equation ay^2=b(x^k-c^k)
    Xiaodong Zhuang, Nikos E. Mastorakis
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. On the Diophantine equation X2+pm=p1n1p2n2
    Amir Ghadermarzi
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. On the Diophantine Equation px+ (2p+1)y =z2 with Consecutive Exponents
    Subhasis Panda
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. On the Diophantine equation Cx2+p2m=yq
    Usha K. Sangale, Priyanka R. Satpute
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{moharana2026matrix,
  title = {{Matrix Solutions to the Diophantine equation $X^4+ Y^4=2Z^4$}},
  author = {Swadhin Moharana and Pabitra Kumar Jena},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32374},
  eprint = {2609.32374},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32374},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Matrix Solutions to the Diophantine equation X^4+ Y^4=2Z^4
AU  - Swadhin Moharana
AU  - Pabitra Kumar Jena
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32374
UR  - https://arxiv.org/abs/2609.32374
AB  - 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(ℤ), M_3(ℤ), M_4(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(ℚ(√[4]l)) and M_4(ℚ(√[4]l_1, √[4]l_2)) when the rational numbers involved lack perfect fourth powers and are not perfect squares.
ER  -