On a conjecture of Levesque and Waldschmidt II
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911857554489344 |
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| author | Hilgart, Tobias Ziegler, Volker |
| author_facet | Hilgart, Tobias Ziegler, Volker |
| contents | Related to Shank's notion of simplest cubic fields, the family of parametrised Diophantine equations,
\[
x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - λ_0 y\right) \left(x-λ_1 y\right) \left(x - λ_2 y\right) = \pm 1,
\]
was studied and solved effectively by Thomas and later solved completely by Mignotte.
An open conjecture of Levesque and Waldschmidt states that taking these parametrised Diophantine equations and twisting them not only once but twice, in the sense that we look at
\[
f_{n,s,t}(x,y) = \left( x - λ_0^s λ_1^t y \right) \left( x - λ_1^sλ_2^t y \right) \left( x - λ_2^sλ_0^t y \right) = \pm 1,
\]
retains a result similar to what Thomas obtained in the original or Levesque and Waldschidt in the once-twisted ($t = 0$) case; namely, that non-trivial solutions can only appear in equations where the parameters are small. We confirm this conjecture, given that the absolute values of the exponents $s, t$ are not too large compared to the base parameter $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Levesque and Waldschmidt II Hilgart, Tobias Ziegler, Volker Number Theory 11D25, 11D57, 11D61 Related to Shank's notion of simplest cubic fields, the family of parametrised Diophantine equations, \[ x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - λ_0 y\right) \left(x-λ_1 y\right) \left(x - λ_2 y\right) = \pm 1, \] was studied and solved effectively by Thomas and later solved completely by Mignotte. An open conjecture of Levesque and Waldschmidt states that taking these parametrised Diophantine equations and twisting them not only once but twice, in the sense that we look at \[ f_{n,s,t}(x,y) = \left( x - λ_0^s λ_1^t y \right) \left( x - λ_1^sλ_2^t y \right) \left( x - λ_2^sλ_0^t y \right) = \pm 1, \] retains a result similar to what Thomas obtained in the original or Levesque and Waldschidt in the once-twisted ($t = 0$) case; namely, that non-trivial solutions can only appear in equations where the parameters are small. We confirm this conjecture, given that the absolute values of the exponents $s, t$ are not too large compared to the base parameter $n$. |
| title | On a conjecture of Levesque and Waldschmidt II |
| topic | Number Theory 11D25, 11D57, 11D61 |
| url | https://arxiv.org/abs/2404.18642 |