Catalan's conjecture

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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[edit]

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[edit]

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 OEISA076427.

See OEISA103953 for the smallest solution (> 0), and OEISA076427 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[edit]

Question dropshade.png 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 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] that there is some positive constant c such that if d is the difference of a perfect power n,[clarification needed] then d>nc for sufficiently large n.

See also[edit]

References[edit]

  1. ^ Victor-Amédée Lebesgue (1850). "Sur l'impossibilité, en nombres entiers, de l'équation xm=y2+1". Nouvelles annales de mathématiques. 1re 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.
  3. ^ a b 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.

External links[edit]