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 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.
for the bound,[4] resolving Catalan's conjecture for all but a finite number of cases.
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.
[6] Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in the OEIS): 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