Catalan's conjecture


Catalan's conjecture is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu. The integers 23 and 32 are two powers of natural numbers whose values are consecutive. The theorem states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of
for a, b > 1, x, y > 0 is x = 3, a = 2, y = 2, b = 3.

History

The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where was restricted to be or. 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 exp exp exp exp 730 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.

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 1018, as per.
See for the smallest solution, and for number of solutions for a given n.
nsolution
count		numbers k such that k and k + n
are both perfect powers		nsolution
count		numbers k such that k and k + n
are both perfect powers
11833216, 256
2125340none
321, 1253531, 289, 1296
434, 32, 12136264, 1728
524, 2737327, 324,
60none3811331
751, 9, 25, 121, 39425, 361, 961,
831, 8, 4049, 81, 216, 2704
9416, 27, 216, 4138, 128, 400
1012187420none
11416, 25, 3125, 3364431441
1224, 219744381, 100, 125
13336, 243, 49004544, 36, 484, 9216
140none461243
1531, 49, 47681, 169, 196, 529, 1681,
1639, 16, 1284841, 16, 121, 21904
1778, 32, 64, 512,,, 49332, 576,
1839, 225, 343500none
1958, 81, 125, 324, 51249, 625
20216, 196521144
2124, 100532676,
22227, 218754227, 289
2344, 9, 121, 20255539, 729,
2451, 8, 25, 1000, 5648, 25, 169, 5776
252100, 14457364, 343, 784
2631,, 580none
2739, 169, 216591841
2874, 8, 36, 100, 484,, 6044, 196,,
29119661264, 900
3016859620none
3121, 2256341, 81, 961,
3244, 32, 49, 774464436, 64, 225, 512

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 has only finitely many solutions with ≠. Pillai proved that the difference 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 of perfect powers satisfies for some positive constant c and all sufficiently large n''.