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Certifying non-existence of integer points on curves

  • Bogdan Grechuk
  • Eugene Go
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Abstract

We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.

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Cite

BibTeXRIS
BibTeX
@misc{grechuk2026certifying,
  title = {{Certifying non-existence of integer points on curves}},
  author = {Bogdan Grechuk and Eugene Go},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18916},
  eprint = {2610.18916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18916},
}
RIS
TY  - UNPB
TI  - Certifying non-existence of integer points on curves
AU  - Bogdan Grechuk
AU  - Eugene Go
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.18916
UR  - https://arxiv.org/abs/2610.18916
AB  - We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.
ER  -