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词条 Catalan's conjecture
释义

  1. History

  2. Generalization

  3. Pillai's conjecture

  4. See also

  5. References

  6. External links

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

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 {{oeis|id=A076427}}.

See {{oeis|id=A103953}} for the smallest solution (> 0), and {{oeis|id=A076427}} for number of solutions (except 0) for a given n.

nsolution
count		numbers k such that k and k + n
are both perfect powers		nsolution
count		numbers k such that k and k + n
are both perfect powers
11833216256
2125340none
32112535312891296
43432121362641728
524273732732414348907
60none3811331
751925121327613942536196110609
8318973364049812162704
941627216640004138128400
1012187420none
114162531253364431441
1224219744381100125
1333624349004544364849216
140none461243
1531491295029476811691965291681250000
16391612848411612121904
177832645127950714060814338415290449332576274576
1839225343500none
19588112532450328435651249625
20216196521144
212410053267624336
22227218754227289
2344912120255539729175561
245182510005429390803125648251695776
25210014457364343784
2631428496436343580none
2739169216591841
287483610048450625131044604419625153962535525316
29119661264900
3016859620none
3121225634181961183250369
3244324977446443664225512

Pillai's conjecture

{{unsolved|mathematics|Does each positive integer occur only finitely many times as a difference of perfect powers?}}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 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, uniformly in m and n.[3]

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

Paul Erdős conjectured {{citation needed|date=May 2017}} that there is some positive constant c such that if d is the difference of a perfect power n,{{clarify|ate=November 2015|date=November 2015}} then d>nc for sufficiently large n.

See also

  • Tijdeman's theorem
  • Størmer's theorem
  • Fermat–Catalan conjecture
  • Beal's conjecture
  • Equation xʸ=yˣ

References

1. ^{{cite journal | author=Victor-Amédée Lebesgue | authorlink=Victor-Amédée Lebesgue | title=Sur l'impossibilité, en nombres entiers, de l'équation xm=y2+1 | journal=Nouvelles annales de mathématiques | series=1re série | volume=9 | year=1850 | pages=178–181 }}
2. ^{{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 }}
3. ^1 {{ 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 }}
4. ^{{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 }}
  • {{cite journal | last=Catalan | first=Eugene | year=1844 | title=Note extraite d’une lettre adressée à l’éditeur | language=fr | journal=J. Reine Angew. Math. | pages=192 | doi=10.1515/crll.1844.27.192 | volume=27 | mr=1578392}}
  • {{cite conference | last=Cohen | first=Henri | year=2005 | title=Démonstration de la conjecture de Catalan | language=fr | trans-title=A proof of the Catalan conjecture | conference=Théorie algorithmique des nombres et équations diophantiennes | publisher=Éditions de l'École Polytechnique | mr=222434 | location=Palaiseau | isbn=2-7302-1293-0 | pages=1–83}}
  • {{cite journal | first=Preda | last=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 | doi=10.1515/crll.2004.048 |mr=2076124}}
  • {{cite book | first=Paulo | last=Ribenboim | authorlink=Paulo Ribenboim | title=Catalan's Conjecture | publisher=Academic Press, Inc. | location=Boston, MA | year=1994 | isbn=0-12-587170-8 | mr=1259738}} Predates Mihăilescu's proof.
  • {{cite journal | first=Robert | last=Tijdeman | authorlink=Robert Tijdeman | title=On the equation of Catalan | journal=Acta Arith. | volume=29 | issue=2 | year=1976 | pages=197–209 | mr=0404137 | doi=10.4064/aa-29-2-197-209}}
  • {{cite journal | first=Tauno | last=Metsänkylä | 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 | mr=2015449| url=http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf }}
  • {{cite journal | first=Yuri | last=Bilu | title=Catalan's conjecture (after Mihăilescu) | journal=Astérisque | volume=294 | year=2004 | pages=vii, 1–26 | nopp=true | mr=2111637 }}

External links

  • {{MathWorld | urlname=CatalansConjecture | title=Catalan's conjecture}}
  • [https://web.archive.org/web/20130122060110/http://www.maa.org/mathland/mathtrek_06_24_02.html Ivars Peterson's MathTrek]
  • On difference of perfect powers
  • Jeanine Daems: [https://web.archive.org/web/20060221125555/http://www.math.leidenuniv.nl/~jdaems/scriptie/Catalan.pdf A Cyclotomic Proof of Catalan's Conjecture]
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4 : Conjectures|Diophantine equations|Theorems in number theory|Conjectures that have been proved
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