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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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 Cremonaโ€™s Elliptic Curve Database to give a wide range of numerical examples.
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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