There are only finitely many irregular Diophantine quadruples

Abstract

A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple {a,b,c,d} with a<b<c<d is called regular if d=a+b+c+2abc+2√((ab+1)(ac+1)(bc+1)). We prove that every irregular Diophantine quadruple satisfies c<10³³⁸ and d<10^10²⁴.

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BibTeX RIS

BibTeX

@misc{he2026there,
  title = {{There are only finitely many irregular Diophantine quadruples}},
  author = {Bo He and Alain Togbe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18912},
  eprint = {2610.18912},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18912},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - There are only finitely many irregular Diophantine quadruples
AU  - Bo He
AU  - Alain Togbe
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.18912
UR  - https://arxiv.org/abs/2610.18912
AB  - A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple {a,b,c,d} with a<b<c<d is called regular if d=a+b+c+2abc+2√((ab+1)(ac+1)(bc+1)). We prove that every irregular Diophantine quadruple satisfies c<10^338 and d<10^10^24.
ER  -

Abstract

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