History
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where x and y were restricted to be 2 or 3.
In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding x,y in terms of a,b to give an effective upper bound for x,y,a,b. Langevin computed a value of exp exp exp exp 730 for the bound. This resolved Catalan's conjecture for all but a finite number of cases. However, the finite calculation required to complete the proof of the theorem was nonetheless too time-consuming to perform.
Catalan's conjecture was proved by Preda Mihăilescu in April 2002, so it is now sometimes called Mihăilescu's theorem. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki.
Read more about this topic: Catalan's Conjecture
Famous quotes containing the word history:
“In history the great moment is, when the savage is just ceasing to be a savage, with all his hairy Pelasgic strength directed on his opening sense of beauty;and you have Pericles and Phidias,and not yet passed over into the Corinthian civility. Everything good in nature and in the world is in that moment of transition, when the swarthy juices still flow plentifully from nature, but their astrigency or acridity is got out by ethics and humanity.”
—Ralph Waldo Emerson (18031882)
“What has history to do with me? Mine is the first and only world! I want to report how I find the world. What others have told me about the world is a very small and incidental part of my experience. I have to judge the world, to measure things.”
—Ludwig Wittgenstein (18891951)
“I am not a literary man.... I am a man of science, and I am interested in that branch of Anthropology which deals with the history of human speech.”
—J.A.H. (James Augustus Henry)