Catalan's conjecture

Catalan's conjecture (occasionally now referred to as 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 conjecture 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 x, a, y, b > 1 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 and y were restricted to be 2 or 3.

In 1976, Robert Tijdeman applied methods from the theory of transcendental numbers to show that there is an effectively computable constant C so that the exponents of all consecutive powers are less than C. As the results of a number of other mathematicians collectively had established a bound for the base dependent only on the exponents, 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 proved 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

Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in OEIS). It states that each positive integer occurs only finitely many times as a difference of perfect powers. It is an open problem and is named for S. S. Pillai.

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.

See also

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

References

• 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. MR2076124. http://www.reference-global.com/doi/abs/10.1515/crll.2004.048. 
• 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. http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf. 
• Yuri Bilu (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque 294: vii, 1–26. 

External links

• Weisstein, Eric W., "Catalan's conjecture" from MathWorld.
• Ivars Peterson's MathTrek
• Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture