The Diophantine equation x/y + y/x + z/w + w/z = n
This paper shows that the equation in the title has no solutions in the positive integers if (i) $n$ has prime factor congruent to $3$ (modulo $4$); or (ii) $8|n$; or $8|n-1$. Case (i) follows from an elementary argument. Case (ii) follows from the previous works of E. Dofs and the second author. Case (iii) follows from a standard Brauer-Manin obstruction argument.
Authors
- Nguyêñ Duy Tân (ORCID: https://orcid.org/0000-0001-9538-1181)
- Nguyen Xuan Tho (ORCID: https://orcid.org/0000-0002-2593-9220)
Publication Details
- Journal
- Proceedings of the Romanian Academy Series A Mathematics Physics Technical Sciences Information Science
- Published
- 2025-09-15
- DOI
- https://doi.org/10.59277/pra-ser.a.26.2.03
- Primary Topic
- Algebraic Geometry and Number Theory
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- National Foundation for Science and Technology Development
- Vietnam Institute for Advanced Study in Mathematics