Skip to content
DraconiticResearch
Search Sources API
186,625 studies31 sourcesUpdated 5 min ago
PreprintarXiv licencearXiv

On a4+b4+c4+d4=(a+27(b+c+d))4 and ten other new infinite families

  • Matej Veselovac
  • Tito Piezas III
Open at sourcePDFDOICite

Abstract

In 2008, Jacobi and Madden proved that a4+b4+c4+d4=e4 has infinitely many primitive solutions with the linear relation e=a+b+c+d. We extend their result to relations with rational cubes e=a+k3(b+c+d) and prove infinitude for k=3 and $10$ other k≠ 1. We compile $30$ starting solutions. In the spirit of Jacobi and Madden's original paper, everything here is simple algebra except one theorem of Mazur.

Related

  1. On infinite knot families with the same HOMFLYPT polynomial
    Yuki Matsushima
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  2. Global log canonical thresholds for Burniat surfaces and some infinite families of surfaces of general type as ℤ2n-covers
    Nguyen Bin, Jheng-Jie Chen, YongJoo Shin
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer
    Yoshichika Iizuka
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Two Infinite Families of Regular Sequences of Power Sums in Three Variables
    Abed Abedelfatah
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. On infinite families of [Pn]-irregular graphs
    Tatiana Dovzhenok, Ilya Lukashenko, Andrei Mikhalev +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{veselovac2026a4b4c4d4a27bcd4,
  title = {{On $a^4+b^4+c^4+d^4=(a+27(b+c+d))^4$ and ten other new infinite families}},
  author = {Matej Veselovac and Tito Piezas III},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12181},
  eprint = {2610.12181},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12181},
}
RIS
TY  - UNPB
TI  - On a^4+b^4+c^4+d^4=(a+27(b+c+d))^4 and ten other new infinite families
AU  - Matej Veselovac
AU  - Tito Piezas III
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12181
UR  - https://arxiv.org/abs/2610.12181
AB  - In 2008, Jacobi and Madden proved that a^4+b^4+c^4+d^4=e^4 has infinitely many primitive solutions with the linear relation e=a+b+c+d. We extend their result to relations with rational cubes e=a+k^3(b+c+d) and prove infinitude for k=3 and $10$ other k≠ 1. We compile $30$ starting solutions. In the spirit of Jacobi and Madden's original paper, everything here is simple algebra except one theorem of Mazur.
ER  -