Catalan's conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 2 and 3 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the "only" case of two consecutive perfect powers. That is to say, that History. The history of the problem dates back at least to Gersonides, who proved a special  case of the conjecture in 1343 where ("x", "y") was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case "b" = 2. 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". Michel Langevin computed a value of formula_1 for the bound. This resolved Catalan's conjecture for all but a finite number of  cases. Nonetheless, the finite calculation required to complete the proof of the theorem was too time-consuming to perform. Catalan's conjecture was proven by Preda Mihăilescu in April 2002. 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. In 2005, Mihăilescu published a simplified proof. Generalization. It is a conjecture that for every natural number "n", there are only finitely many pairs of perfect powers with difference  n. The list below shows, for "n" ≤ 64, all solutions for perfect powers less than 10, as . See also for the smallest solution (> 0). Pillai's conjecture. Pillai's conjecture concerns a general difference of perfect powers : it is an open problem initially proposed by S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity. This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers "A", "B", "C" the equation  formula_2 has only finitely many solutions ("x", "y", "m", "n") with ("m", "n") ≠ (2, 2). Pillai proved that the difference formula_3 for any λ less than 1, uniformly in "m" and "n". The general conjecture would follow from the ABC conjecture. Paul Erdős conjectured that the ascending sequence formula_4 of perfect powers satisfies formula_5 for some positive constant "c" and all sufficiently large "n".