## A modular approach to cubic Thue-Mahler equations

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- by Dohyeong Kim PDF
- Math. Comp.
**86**(2017), 1435-1471 Request permission

## Abstract:

Let $h(x,y)$ be a non-degenerate binary cubic form with integral coefficients, and let $S$ be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers $x,y$ such that $h(x,y)$ is an $S$-unit. In the present paper, we reverse a well-known argument, which seems to go back to Shafarevich, and use the modularity of elliptic curves over $\mathbb {Q}$ to give upper bounds for the number of solutions of such a Thue-Mahler equation. In addition, our methods give an effective method for determining all solutions, and we use Cremonaโs Elliptic Curve Database to give a wide range of numerical examples.## References

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## Additional Information

**Dohyeong Kim**- Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), 77 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 790-784, Republic of Korea โ and โ Pohang University of Science and Technology (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 790-784, Republic of Korea
- Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, Ann Arbor, Michigan 48109-1043
- MR Author ID: 970842
- Email: dohyeong@umich.edu
- Received by editor(s): June 9, 2015
- Received by editor(s) in revised form: November 27, 2015
- Published electronically: September 15, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 1435-1471 - MSC (2010): Primary 11D59, 11F11, 11Y50
- DOI: https://doi.org/10.1090/mcom/3139
- MathSciNet review: 3614023