Catalan's conjecture

For Catalan's aliquot sequence conjecture, see aliquot sequence.

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. The integers 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). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case b = 2.^[1]

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.^[2] 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 A076427.

See A103953 for the smallest solution (> 0), and A076427 for number of solutions (except 0) for a given n.

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

Pillai's conjecture

Unsolved problem in mathematics:

Does each positive integer occur only finitely many times as a difference of perfect powers?

(more unsolved problems in mathematics)

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 $Ax^{n}-By^{m}=C$ 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, uniformly in m and n.^[3]

The general conjecture would follow from the ABC conjecture.^[3]^[4]

Paul Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n, then d>n^c for sufficiently large n.

References

↑ Victor-Amédée Lebesgue (1850). "Sur l'impossibilité, en nombres entiers, de l'équation x^m=y²+1". Nouvelles annales de mathématiques. 1^re série. 9: 178–181.
↑ Ribenboim, Paulo (1979). 13 Lectures on Fermat's Last Theorem. Springer-Verlag. p. 236. ISBN 0-387-90432-8. Zbl 0456.10006.
1 2 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. (in French). 27: 192. doi:10.1515/crll.1844.27.192. MR 1578392.
Cohen, Henri (2005). Démonstration de la conjecture de Catalan [A proof of the Catalan conjecture]. Théorie algorithmique des nombres et équations diophantiennes (in French). Palaiseau: Éditions de l'École Polytechnique. pp. 1–83. ISBN 2-7302-1293-0. MR 0222434.
Mihăilescu, Preda (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 572: 167–195. doi:10.1515/crll.2004.048. MR 2076124.
Ribenboim, Paulo (1994). Catalan's Conjecture. Boston, MA: Academic Press, Inc. ISBN 0-12-587170-8. MR 1259738. Predates Mihăilescu's proof.
Tijdeman, Robert (1976). "On the equation of Catalan". Acta Arith. 29 (2): 197–209. doi:10.4064/aa-29-2-197-209. MR 0404137.
Metsänkylä, Tauno (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. MR 2015449.
Bilu, Yuri (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque. 294: vii, 1–26. MR 2111637.

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

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[1] Victor-Amédée Lebesgue (1850). "Sur l'impossibilité, en nombres entiers, de l'équation x^m=y²+1". Nouvelles annales de mathématiques. 1^re série. 9: 178–181.

[2] Ribenboim, Paulo (1979). 13 Lectures on Fermat's Last Theorem. Springer-Verlag. p. 236. ISBN 0-387-90432-8. Zbl 0456.10006.

[rnt-3] 1 2 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.

[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.