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:
“No cause is left but the most ancient of all, the one, in fact, that from the beginning of our history has determined the very existence of politics, the cause of freedom versus tyranny.”
—Hannah Arendt (1906–1975)
“The history of modern art is also the history of the progressive loss of art’s audience. Art has increasingly become the concern of the artist and the bafflement of the public.”
—Henry Geldzahler (1935–1994)
“The history of progress is written in the blood of men and women who have dared to espouse an unpopular cause, as, for instance, the black man’s right to his body, or woman’s right to her soul.”
—Emma Goldman (1869–1940)