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Baker’s Method and Tijdeman’s Argument

In: The Problem of Catalan

Author

Listed:
  • Yuri F. Bilu

    (University of Bordeaux and CNRS, Institute of Mathematics of Bordeaux)

  • Yann Bugeaud

    (University of Strasbourg and CNRS, IRMA, Mathematical Institute)

  • Maurice Mignotte

    (University of Strasbourg and CNRS, IRMA, Mathematical Institute)

Abstract

This chapter is somewhat isolated and can be read (almost) independently of the others. We discuss in it the application of Baker’s method to Diophantine equations of Catalan type. We give a brief introduction to this method, show how it applies to classical Diophantine equations, and reproduce the beautiful argument of Tijdeman, who proved that Catalan’s equation has only finitely many solutions. Moreover, the solutions are bounded by an absolute effective constant (that is, a constant that can, in principle, be explicitly determined), which reduces the problem to a finite computation. Before the work of Mihăilescu this was the top achievement on Catalan’s problem.We also consider the more general equation of Pillai and show that it has finitely many solutions when one of the four variables is fixed.

Suggested Citation

  • Yuri F. Bilu & Yann Bugeaud & Maurice Mignotte, 2014. "Baker’s Method and Tijdeman’s Argument," Springer Books, in: The Problem of Catalan, edition 127, chapter 0, pages 159-233, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-10094-4_13
    DOI: 10.1007/978-3-319-10094-4_13
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