The Real Part of Mihăilescu’s Ideal

Yuri F. Bilu, Yann Bugeaud and Maurice Mignotte
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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

Chapter Chapter 9 in The Problem of Catalan, 2014, pp 117-128 from Springer

Abstract: Abstract In this chapter we continue our study of Mihăilescu’s ideal. As follows from the definition, it contains the ideal q ℤ [ G ] $$q\mathbb{Z}[G]$$ of the elements divisible by q. A basic question is whether Mihăilescu’s ideal has nontrivial (that is, not divisible by q) elements.In this chapter we prove that (for large x) the real part ℐ M + $$\mathcal{I}_{M}^{+}$$ of Mihăilescu’s ideal contains no nontrivial elements of weight 0 and even of any weight divisible by q.On the other hand, later we shall see that a solution to Catalan’s equation gives rise to a nontrivial element of ℐ M + $$\mathcal{I}_{M}^{+}$$ of weight divisible by q. This contradiction would prove Catalan’s conjecture.

Keywords: Nontrivial Element; Binomial Power Series; Algebraic Integers; Nonnegative Real Coefficients; Similar General Conditions (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-10094-4_9

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DOI: 10.1007/978-3-319-10094-4_9

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