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'''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 [[perfect power|power]]s of [[natural number]]s, 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 [[Diophantine equation|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 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.<ref>{{cite book | title=13 Lectures on Fermat's Last Theorem | first=Paulo | last=Ribenboim | authorlink=Paulo Ribenboim | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90432-8 | zbl=0456.10006 | page=236 }}</ref> 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 field]]s and [[Galois module]]s. 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 {{OEIS|id=A001597}}: 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 <math>Ax^n - By^m = C</math> has only finitely many solutions (''x'',''y'',''m'',''n'') with (''m'',''n'') ≠ (2,2). Pillai proved that the difference <math>|Ax^n - By^m| \gg x^{\lambda n}</math> for any λ less than 1.<ref name=rnt>{{ cite book | pages=253–254 | title=Rational Number Theory in the 20th Century: From PNT to FLT | series=Springer Monographs in Mathematics | first=Wladyslaw | last=Narkiewicz | publisher=[[Springer-Verlag]] | year=2011 | isbn=0-857-29531-4 }}</ref>

The general conjecture would follow from the [[ABC conjecture]].<ref name=rnt/><ref>{{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 | page=207 }}</ref>

[[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==

{{reflist}}
* Catalan, Eugene. (1844): {{lang|fr|Note extraite d’une lettre adressée à l’éditeur. J. Reine Angew. Math., 27:192.}}
* {{cite journal | author=Preda Mihăilescu | authorlink=Preda Mihăilescu | title=Primary Cyclotomic Units and a Proof of Catalan's Conjecture | journal=J. Reine angew. Math. | volume=572 | year=2004 | pages=167–195 | url=http://www.reference-global.com/doi/abs/10.1515/crll.2004.048 | doi=10.1515/crll.2004.048 | issue=572 |mr=2076124}}
* {{cite book | author=Paulo Ribenboim | authorlink=Paulo Ribenboim | title=Catalan's Conjecture | publisher=Academic Press | year=1994 | isbn=0-12-587170-8 }} Predates Mihăilescu's proof.
* {{cite journal | author=Robert Tijdeman | authorlink=Robert Tijdeman | title=On the equation of Catalan | journal=Acta Arith. | volume=29 | issue=2 | year=1976 | pages=197–209 }}
* {{cite journal | author=Tauno Metsänkylä | url=http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf | format=PDF | title=Catalan's conjecture: another old Diophantine problem solved | journal=[[Bulletin of the American Mathematical Society]] | volume=41 | year=2004 | issue=1 | pages=43–57 | doi=10.1090/S0273-0979-03-00993-5 }}
* {{cite journal | author=Yuri Bilu | title=Catalan's conjecture (after Mihăilescu) | journal=[[Astérisque]] | volume=294 | year=2004 | pages=vii, 1–26 | nopp=true }}

==External links==
* {{MathWorld | urlname=CatalansConjecture | title=Catalan's conjecture}}
* [http://www.maa.org/mathland/mathtrek_06_24_02.html Ivars Peterson's MathTrek]
* Jeanine Daems: [http://www.math.leidenuniv.nl/~jdaems/scriptie/Catalan.pdf A Cyclotomic Proof of Catalan's Conjecture]

[[Category:Conjectures]]
[[Category:Diophantine equations]]
[[Category:Theorems in number theory]]

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en>Wikiain: Reverted 1 edit by 151.227.113.48 (talk) to last revision by Delusion23. (TW)