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.

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

xa − yb = 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, 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.

Generalization

It is a conjecture that for every natural number n, there are only finitely many pairs of perfect powers with difference n; see the list.

(For the smallest number (>0), see  A103953, and see  A076427 for number of solutions (except 0))

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

Pillai's conjecture

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

See also

References

  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. 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.

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