Glasnik Matematicki, Vol. 60, No. 1 (2025), 1-19. \( \)

ON A CONJECTURE OF LEVESQUE AND WALDSCHMIDT

Tobias Hilgart and Volker Ziegler

Department of Mathematics, University of Salzburg, 5020 Salzburg, Austria
e-mail:tobias.hilgart@plus.ac.at

Department of Mathematics, University of Salzburg, 5020 Salzburg, Austria
e-mail:volker.ziegler@plus.ac.at

Abstract.   One of the first parametrised Thue equations, \[ \hspace{-18ex}\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1, \] over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as \[ \hspace{-18ex} \left| N_{K/\mathbb{Q}}\left( X - \lambda_0 Y \right) \right| = \left| \left( X-\lambda_0 Y \right) \left( X-\lambda_1 Y \right) \left( X-\lambda_2 Y \right) \right| =1 \] if \(\lambda_0, \lambda_1, \lambda_2\) are the roots of the defining irreducible polynomial, and \(K\) is the corresponding number field. Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter \(s\) and looked, among other things, at the equation \[ \hspace{-18ex} \left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s Y \right) \right| = 1. \] They solved this effectively and conjectured that introducing a second exponential parameter \(t\) and looking at \(\left| N_{K/\mathbb{Q}}\left( X - \lambda_0^s\lambda_1^t Y \right) \right| = 1\) does not change the effective solvability. We want to partially confirm this if \[ \hspace{-18ex} \min\left\{ \left| 2s-t \right|, \left| 2t-s \right|, \left| s+t \right| \right\} \gt \varepsilon \cdot \max\left\{ \left|s\right|, \left|t\right| \right\} \gt 2, \] i.e., the two exponents do not almost cancel in specific cases.

2020 Mathematics Subject Classification.   11D25, 11D57

Key words and phrases.   Parametrised Thue equations, exponential Diophantine equations

https://doi.org/10.3336/gm.60.1.01

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