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.

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

x^a − y^b = 1

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 (x, y) was restricted to be (2, 3) or (3, 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. Langevin computed a value of exp exp exp exp 730 for the bound.[1] 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.
Generalization

It is a conjecture that for every natural number n, there are only finite pairs of perfect powers with difference n, see the list.

(For the smallest number (>0), see OEIS A103953, and see OEIS A076427 for number of solutions (except 0))

n	numbers k such that k and k + n are both perfect powers	n	numbers k such that k and k + n are both perfect powers
1	0, 8	33	16, 256
2	25	34	None
3	1, 125	35	1, 289, 1296
4	0, 4, 32, 121	36	0, 64, 1728
5	4, 27	37	27, 324, 14348907
6	None	38	1331
7	1, 9, 25, 121, 32761	39	25, 361, 961, 10609
8	0, 1, 8, 97336	40	9, 81, 216, 2704
9	0, 16, 27, 216, 64000	41	8, 128, 400
10	2187	42	None
11	16, 25, 3125, 3364	43	441
12	4, 2197	44	81, 100, 125
13	36, 243, 4900	45	4, 36, 484, 9216
14	None	46	243
15	1, 49, 1295029	47	81, 169, 196, 529, 1681, 250000
16	0, 9, 16, 128	48	1, 16, 121, 21904
17	8, 32, 64, 512, 79507, 140608, 143384152904	49	0, 32, 576, 274576
18	9, 225, 343	50	None
19	8, 81, 125, 324, 503284356	51	49, 625
20	16, 196	52	144
21	4, 100	53	676, 24336
22	27, 2187	54	27, 289
23	4, 9, 121, 2025	55	9, 729, 175561
24	1, 8, 25, 1000, 542939080312	56	8, 25, 169, 5776
25	0, 100, 144	57	64, 343, 784
26	1, 42849, 6436343	58	None
27	0, 9, 169, 216	59	841
28	4, 8, 36, 100, 484, 50625, 131044	60	4, 196, 2515396, 2535525316
29	196	61	64, 900
30	6859	62	None
31	1, 225	63	1, 81, 961, 183250369
32	0, 4, 32, 49, 7744	64	0, 36, 64, 225, 512

Pillai's conjecture

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 \( Ax^n - By^m = C \) has only finitely many solutions (x,y,m,n) with (m,n) ≠ (2,2). Pillai proved that the difference \( |Ax^n - By^m| \gg x^{\lambda n} \) for any λ less than 1.[2]

The general conjecture would follow from the ABC conjecture.[2][3]

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.

See also

Tijdeman's theorem
Størmer's theorem
Fermat–Catalan conjecture
Beal's conjecture

References

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.

External links

Weisstein, Eric W., "Catalan's conjecture", MathWorld.
Ivars Peterson's MathTrek
On difference of perfect powers
Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture

Mathematics Encyclopedia