On a family of unit equations over simplest cubic fields
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 705-718.

Let $a\in ℤ$ and $\rho$ be a root of ${f}_{a}\left(x\right)={x}^{3}-a{x}^{2}-\left(a+3\right)x-1$, then the number field ${K}_{a}=ℚ\left(\rho \right)$ is called a simplest cubic field. In this paper we consider the family of unit equations ${u}_{1}+{u}_{2}=n$ where ${u}_{1},{u}_{2}\in ℤ{\left[\rho \right]}^{*}$ and $n\in ℤ$. We completely solve the unit equations under the restriction $|n|\le max\left\{1,|a{|}^{1/3}\right\}$.

Soient $a\in ℤ$ et $\rho$ une racine de ${f}_{a}\left(x\right)={x}^{3}-a{x}^{2}-\left(a+3\right)x-1$. On dit que le corps de nombres ${K}_{a}=ℚ\left(\rho \right)$ est un corps cubique élémentaire. Dans cet article, nous considérons la famille des équations en unités algébriques de la forme ${u}_{1}+{u}_{2}=n$${u}_{1},{u}_{2}\in ℤ{\left[\rho \right]}^{*}$ et $n\in ℤ$. Nous résolvons complètement ces équations sous l’hypothèse $|n|\le max\left\{1,{|a|}^{1/3}\right\}$.

Accepted:
Published online:
DOI: 10.5802/jtnb.1223
Classification: 11D61, 11D75, 11J86
Keywords: Diophantine equations, unit equations, simplest cubic fields
Ingrid Vukusic 1; Volker Ziegler 1

1 University of Salzburg Hellbrunnerstrasse 34/I A-5020 Salzburg, Austria
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Ingrid Vukusic; Volker Ziegler. On a family of unit equations over simplest cubic fields. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 705-718. doi : 10.5802/jtnb.1223. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1223/

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