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PreprintCC BY 4.0arXiv

Diophantine Equation x13βˆ’x22x1+1=0 over number fields

  • Pinki Khatun
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Abstract

Let 𝕂=β„š(√(βˆ’d)) or β„š(√d), where d is a positive square-free integer, and denote by π’ͺK the ring of integers of 𝕂. I investigate the solution of the equation x13βˆ’x22x1+1=0 where x1∈ 𝕂 and x2∈π’ͺK. The case 𝕂=β„š(√d) for d≑ 2,3pmod4 faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings β„€[√d]], together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely (x1,x2)=(βˆ’1,0) and (1,√2) for 𝕂=β„š(√2) and one solution (βˆ’1,0) for 𝕂=β„š(√d) As an application, I consider the family of elliptic curves Cm:Y2=X3βˆ’m2X+1, m∈π’ͺK, and deduce that, for every mβ‰ 0,√2 the Mordell--Weil group Cm(𝕂) contains no rational point of order two.

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Cite

BibTeXRIS
BibTeX
@misc{khatun2026diophantine,
  title = {{Diophantine Equation $x_1^3-x_2^2x_1+1=0$ over number fields}},
  author = {Pinki Khatun},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.26079},
  eprint = {2607.26079},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.26079},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Diophantine Equation x_1^3βˆ’x_2^2x_1+1=0 over number fields
AU  - Pinki Khatun
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2607.26079
UR  - https://arxiv.org/abs/2607.26079
AB  - Let 𝕂=β„š(√(βˆ’d)) or β„š(√d), where d is a positive square-free integer, and denote by π’ͺ_K the ring of integers of 𝕂. I investigate the solution of the equation x_1^3βˆ’x_2^2x_1+1=0 where x_1∈ 𝕂 and x_2∈π’ͺ_K. The case 𝕂=β„š(√d) for d≑ 2,3pmod4 faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings β„€[√d]], together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely (x_1,x_2)=(βˆ’1,0) and (1,√2) for 𝕂=β„š(√2) and one solution (βˆ’1,0) for 𝕂=β„š(√d) As an application, I consider the family of elliptic curves C_m:Y^2=X^3βˆ’m^2X+1, m∈π’ͺ_K, and deduce that, for every mβ‰ 0,√2 the Mordell--Weil group C_m(𝕂) contains no rational point of order two.
ER  -