Catalan's conjecture
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
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Unsolved problem in mathematics: Does each positive integer occur only finitely many times as a difference of perfect powers? (more unsolved problems in mathematics)
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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
- Beal's conjecture
- Equation xy = yx
- Fermat–Catalan conjecture
- Mordell curve
- Ramanujan–Nagell equation
- Størmer's theorem
- Tijdeman's theorem
- Thaine's theorem
Notes
- ↑ Weisstein, Eric W., Catalan's conjecture, MathWorld, https://mathworld.wolfram.com/CatalansConjecture.html
- ↑ Mihăilescu 2004
- ↑ 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
- ↑ Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 0-387-90432-8
- ↑ 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/
- ↑ Mihăilescu 2005
- ↑ 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
- ↑ 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
- Bilu, Yuri (2004), "Catalan's conjecture (after Mihăilescu)", Astérisque 294: vii, 1–26
- Catalan, Eugene (1844), "Note extraite d'une lettre adressée à l'éditeur" (in fr), J. Reine Angew. Math. 27: 192, doi:10.1515/crll.1844.27.192, https://zenodo.org/record/1448842
- Cohen, Henri (2005). "Démonstration de la conjecture de Catalan" (in fr). Théorie algorithmique des nombres et équations diophantiennes. Palaiseau: Éditions de l'École Polytechnique. pp. 1–83. ISBN 2-7302-1293-0.
- Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved", Bulletin of the American Mathematical Society 41 (1): 43–57, doi:10.1090/S0273-0979-03-00993-5, https://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf
- Mihăilescu, Preda (2004), "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", J. Reine Angew. Math. 2004 (572): 167–195, doi:10.1515/crll.2004.048
- Mihăilescu, Preda (2005), "Reflection, Bernoulli numbers and the proof of Catalan's conjecture", European Congress of Mathematics (Zurich: Eur. Math. Soc.): 325-340, https://www.uni-math.gwdg.de/preda/mihailescu-papers/catber.pdf
- Ribenboim, Paulo (1994), Catalan's Conjecture, Boston, MA: Academic Press, Inc., ISBN 0-12-587170-8 Predates Mihăilescu's proof.
- Tijdeman, Robert (1976), "On the equation of Catalan", Acta Arith. 29 (2): 197–209, doi:10.4064/aa-29-2-197-209, https://www.impan.pl/shop/publication/transaction/download/product/100989?download.pdf
External links
- Weisstein, Eric W.. "Catalan's conjecture". http://mathworld.wolfram.com/CatalansConjecture.html.
- Ivars Peterson's MathTrek
- On difference of perfect powers
- Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture
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