23 and 32 are two powers of natural numbers, whose values 8 and 9 respectively 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
- xa − yb = 1
for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3.
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 proven 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.
Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in 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 has only finitely many solutions (x,y,m,n) with (m,n) ≠ (2,2). Pillai proved that the difference for any λ less than 1.
Paul Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n, then d>nc for sufficiently large n.
- Ribenboim, Paulo (1979). 13 Lectures on Fermat's Last Theorem. Springer-Verlag. p. 236. ISBN 0-387-90432-8. Zbl 0456.10006.
- Narkiewicz, Wladyslaw (2011). Rational Number Theory in the 20th Century: From PNT to FLT. Springer Monographs in Mathematics. Springer-Verlag. pp. 253–254. ISBN 0-857-29531-4.
- Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics 1467 (2nd ed.). Springer-Verlag. p. 207. ISBN 3-540-54058-X. Zbl 0754.11020.
- Catalan, Eugene. (1844): Note extraite d’une lettre adressée à l’éditeur. J. Reine Angew. Math., 27:192.
- Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine angew. Math. 572 (572): 167–195. doi:10.1515/crll.2004.048. MR 2076124.
- Paulo Ribenboim (1994). Catalan's Conjecture. Academic Press. ISBN 0-12-587170-8. Predates Mihăilescu's proof.
- Robert Tijdeman (1976). "On the equation of Catalan". Acta Arith. 29 (2): 197–209.
- Tauno Metsänkylä (2004). "Catalan's conjecture: another old Diophantine problem solved" (PDF). Bulletin of the American Mathematical Society 41 (1): 43–57. doi:10.1090/S0273-0979-03-00993-5.
- Yuri Bilu (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque 294: vii, 1–26.