Catalan's conjecture

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Short description: The only nontrivial positive integer solution to x^a-y^b equals 1 is 3^2-2^3


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 at Paderborn University.[1][2] The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that

Catalan's conjecture — the only solution in the natural numbers of

[math]\displaystyle{ x^a - y^b = 1 }[/math]

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.[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. Michel Langevin computed a value of [math]\displaystyle{ \exp \exp \exp \exp 730 \approx 10^{10^{10^{10^{317}}}} }[/math] for the bound,[4] resolving Catalan's conjecture for all but a finite number of cases.

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.[5] In 2005, Mihăilescu published a simplified proof.[6]

Pillai's conjecture

Question, Web Fundamentals.svg 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 [math]\displaystyle{ 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]\displaystyle{ |Ax^n - By^m| \gg x^{\lambda n} }[/math] for any λ less than 1, uniformly in m and n.[7]

The general conjecture would follow from the ABC conjecture.[7][8]

Paul Erdős conjectured[citation needed] that the ascending sequence [math]\displaystyle{ (a_n)_{n\in\mathbb N} }[/math] of perfect powers satisfies [math]\displaystyle{ a_{n+1} - a_n \gt n^c }[/math] for some positive constant c and all sufficiently large n.

Pillai's conjecture means 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 OEIS: A076427. See also OEIS: A103953 for the smallest solution (> 0).

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

See also

Notes

  1. ↑ Weisstein, Eric W., Catalan's conjecture, MathWorld, https://mathworld.wolfram.com/CatalansConjecture.html 
  2. ↑ Mihăilescu 2004
  3. ↑ 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 
  4. ↑ Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 0-387-90432-8 
  5. ↑ Bilu, Yuri (2004), "Catalan's conjecture", Séminaire Bourbaki vol. 2003/04 Exposés 909-923, Astérisque, 294, pp. 1–26, http://www.numdam.org/book-part/SB_2002-2003__45__1_0/ 
  6. ↑ Mihăilescu 2005
  7. ↑ 7.0 7.1 Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, pp. 253–254, ISBN 978-0-857-29531-6, https://archive.org/details/rationalnumberth00nark 
  8. ↑ 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 

References

External links



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