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On the representation of unity by binary cubic forms


Author: Michael A. Bennett
Journal: Trans. Amer. Math. Soc. 353 (2001), 1507-1534
MSC (2000): Primary 11D25; Secondary 11E76
DOI: https://doi.org/10.1090/S0002-9947-00-02658-1
Published electronically: December 18, 2000
MathSciNet review: 1806730
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Abstract:

If $F(x,y)$ is a binary cubic form with integer coefficients such that $F(x,1)$has at least two distinct complex roots, then the equation $F(x,y) = 1$possesses at most ten solutions in integers $x$ and $y$, nine if $F$ has a nontrivial automorphism group. If, further, $F(x,y)$ is reducible over $\mathbb{Z}[x,y]$, then this equation has at most $2$ solutions, unless $F(x,y)$ is equivalent under $GL_2(\mathbb{Z})$-action to either $x (x^2-xy-y^2)$ or $x (x^2-2y^2)$. The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations $F(x,y)=1$ for $F$ cubic and irreducible of positive discriminant $D_F \leq 10^6$. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form $F(x,y)=m$ and to Mordell's equation $y^2=x^3+k$, where $m$ and $k$ are nonzero integers.


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Additional Information

Michael A. Bennett
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Champaign, Illinois 61801
Email: mabennet@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02658-1
Keywords: Thue equations, binary cubic forms
Received by editor(s): October 21, 1999
Received by editor(s) in revised form: December 8, 1999
Published electronically: December 18, 2000
Additional Notes: The author was supported in part by NSF Grant DMS-9700837
Article copyright: © Copyright 2000 American Mathematical Society

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