Titlebook: Catalan‘s Conjecture; René Schoof Textbook 2008 Springer-Verlag London 2008 Algebra.Arithmetic.Catalan‘s conjecture.algebraic number theor

蝕刻術(shù)

The Nontrivial Solution,In this chapter, we present a different solution. It is due to W. McCallum [31]. He reduces the problem to a Thue equation, which he solves by means of Skolem’s method. See [9, s. 10.10] for this method.

Corral

Cloudburst

An Obstruction Group,Nonzero solutions to Catalan’s equation . naturally give rise to elements in certain obstruction groups. Mih?ilescu’s theorems can be viewed as results about the size of the submodules generated by these elements. In this section, we introduce the notions and notations that are used in the remaining sections.

不幸的人

Small , or ,,In this chapter, we present Mih?ilescu’s proof of Theorem IV mentioned in chapter 1. The proof is by a .-adic argument. Let . be distinct odd primes.

輕推

The Stickelberger Ideal,Let . be an odd prime and let .. denote a primitive .th root of unity.

或者發(fā)神韻

Gossamer

The Minus Argument,In this chapter, we prove Theorem III of chapter 1. The proof is “Archimedean” in the sense that it exploits the fact that for any nonzero solution .,. of Catalan’s equation ., the absolute values of . and . are necessarily very large. This follows from Corollary 6.5 (iii). We use the notation introduced in chapter 7.

檢查

The Plus Argument I,In this chapter and chapter 14, we prove Theorem II of chapter 1. There are two main ingredients: the Runge method, exploited here, and Thaine’s theorem, which is used in chapter 14.

無關(guān)緊要

metropolitan

The Plus Argument II,In this chapter, we prove Theorem II of chapter 1. The key ingredients are the results of chapter 12 and Francisco Thaine’s famous theorem [48], which is proved in chapter 16. Let . be odd primes and let . be a nonzero solution to Catalan’s equation ..