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:
“If you look at the 150 years of modern China’s history since the Opium Wars, then you can’t avoid the conclusion that the last 15 years are the best 15 years in China’s modern history.”
—J. Stapleton Roy (b. 1935)
“History does nothing; it does not possess immense riches, it does not fight battles. It is men, real, living, who do all this.... It is not “history” which uses men as a means of achieving—as if it were an individual person—its own ends. History is nothing but the activity of men in pursuit of their ends.”
—Karl Marx (1818–1883)
“The history of literature—take the net result of Tiraboshi, Warton, or Schlegel,—is a sum of a very few ideas, and of very few original tales,—all the rest being variation of these.”
—Ralph Waldo Emerson (1803–1882)