Binomial Thue equations and polynomial powers

M. A. Bennett, K. Gyory, M. Mignotte, Á Pintér

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

We explicitly solve a collection of binomial Thue equations with unknown degree and unknown S-unit coefficients, for a number of sets S of small cardinality. Equivalently, we characterize integers x such that the polynomial x2 + x assumes perfect power values, modulo S-units. These results are proved through a combination of techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, and computational approaches to Thue equations of low degree. Along the way, we derive some new results on Fermat-type ternary equations, combining classical cyclotomy with Frey curve techniques.

Original languageEnglish
Pages (from-to)1103-1121
Number of pages19
JournalCompositio Mathematica
Volume142
Issue number5
DOIs
Publication statusPublished - 2006

Keywords

  • Binomial Thue equations
  • Explicit resolution
  • Superelliptic equations

ASJC Scopus subject areas

  • Algebra and Number Theory

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