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 as in human life, regret does not bring back a lost moment and a thousand years will not recover something lost in a single hour.”
—Stefan Zweig (18811942)
“Systematic philosophical and practical anti-intellectualism such as we are witnessing appears to be something truly novel in the history of human culture.”
—Johan Huizinga (1872–1945)
“All objects, all phases of culture are alive. They have voices. They speak of their history and interrelatedness. And they are all talking at once!”
—Camille Paglia (b. 1947)