Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 41 min ago
PreprintCC BY 4.0arXiv

Some Methods to Solve the Diophantine Equation ay^2=b(x^k-c^k)

  • Xiaodong Zhuang
  • Nikos E. Mastorakis
Open at sourcePDFDOICite

Abstract

The solution of the Diophantine equation ay^2=b(x^k-c^k) is studied in different cases of k. Methods are proposed for different cases of k, which are convenient for computer algorithm implementation. The original equation is reduced to lower-degree equations, which can improve the efficiency in searching possible integer solutions. A sufficient condition of the unsolvability of ay^2=b(x^2n-c^2n) is presented.

Related

  1. The Diophantine Problem with constraints in some free-by-cyclic groups
    Laura Ciobanu, Robert D. Gray, Alex Levine
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. An existence result of a functional integral equation via Darbo type theorem and an iterative algorithm to solve it
    Bhuban Chandra Deuri, Anupam Das
    arXivPreprintarXiv licenceFull textAnalysis & PDEsNumerical analysis & scientific computingarXiv
  3. When does least-squares residual minimization solve a differential equation?
    Carlos Esteve-Yagüe, Richard Tsai, Stanley Osher
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  4. On the Diophantine equation X2+pm=p1n1p2n2
    Amir Ghadermarzi
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. On the Diophantine Equation px+ (2p+1)y =z2 with Consecutive Exponents
    Subhasis Panda
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{zhuang2026some,
  title = {{Some Methods to Solve the Diophantine Equation ay^2=b(x^k-c^k)}},
  author = {Xiaodong Zhuang and Nikos E. Mastorakis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00119},
  eprint = {2610.00119},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00119},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Some Methods to Solve the Diophantine Equation ay^2=b(x^k-c^k)
AU  - Xiaodong Zhuang
AU  - Nikos E. Mastorakis
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00119
UR  - https://arxiv.org/abs/2610.00119
AB  - The solution of the Diophantine equation ay^2=b(x^k-c^k) is studied in different cases of k. Methods are proposed for different cases of k, which are convenient for computer algorithm implementation. The original equation is reduced to lower-degree equations, which can improve the efficiency in searching possible integer solutions. A sufficient condition of the unsolvability of ay^2=b(x^2n-c^2n) is presented.
ER  -