# 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 e 32 are two perfect powers (questo è, powers of exponent higher than one) of natural numbers whose values (8 e 9, rispettivamente) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. Vale a dire, that Catalan's conjecture — the only solution in the natural numbers of {stile di visualizzazione x^{un}-si^{b}=1} for a, b > 1, X, y > 0 is x = 3, a = 2, y = 2, b = 3.

Contenuti 1 Storia 2 Pillai's conjecture 3 Guarda anche 4 Appunti 5 Riferimenti 6 External links History The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 dove (X, y) was restricted to be (2, 3) o (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,un,b. Michel Langevin computed a value of {displaystyle exp exp exp exp 730approx 10^{10^{10^{10^{317}}}}} 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 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: più generalmente, in 1931 Pillai conjectured that for fixed positive integers A, B, C the equation {displaystyle Ax^{n}-By^{m}=C} has only finitely many solutions (X, y, m, n) insieme a (m, n) ≠ (2, 2). Pillai proved that the difference {stile di visualizzazione |Ax^{n}-By^{m}|gg x^{lambda n}} 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[citazione necessaria] that the ascending sequence {stile di visualizzazione (un_{n})_{nin mathbb {N} }} of perfect powers satisfies {stile di visualizzazione a_{n+1}-un_{n}>n^{c}} 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 nessuno 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 nessuno 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 nessuno 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 nessuno 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 nessuno 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 nessuno 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 nessuno 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 Notes ^ Weisstein, Eric W., Catalan's conjecture, MathWorld ^ Mihăilescu 2004 ^ Victor-Amédée Lebesgue (1850), "Sur l'impossibilité, en nombres entiers, de l'équation xm=y2+1", Nuovi Annali di Matematica, 1re série, 9: 178–181 ^ Ribenboim, Paulo (1979), 13 Lezioni sull'ultimo teorema di Fermat, Springer-Verlag, p. 236, ISBN 0-387-90432-8, Zbl 0456.10006 ^ Bilu, Yuri (2004), "Catalan's conjecture", Séminaire Bourbaki vol. 2003/04 Exposés 909-923, Asterisco, vol. 294, pp. 1–26 ^ Mihăilescu 2005 ^ Salta su: a b Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Monografie di Springer in matematica, Springer-Verlag, pp. 253–254, ISBN 978-0-857-29531-6 ^ Schmidt, Wolfgang M. (1996), Diophantine approximations and Diophantine equations, Appunti delle lezioni in matematica, vol. 1467 (2nd ed.), Springer-Verlag, p. 207, ISBN 3-540-54058-X, Zbl 0754.11020 References Bilu, Yuri (2004), "Catalan's conjecture (after Mihăilescu)", Asterisco, 294: vii, 1–26, SIG 2111637 Catalan, Eugenio (1844), "Note extraite d'une lettre adressée à l'éditeur", J. Reine Angew. Matematica. (in francese), 27: 192, doi:10.1515/crll.1844.27.192, SIG 1578392 Cohen, Henri (2005). Démonstration de la conjecture de Catalan [A proof of the Catalan conjecture]. Théorie algorithmique des nombres et équations diophantiennes (in francese). Palaiseau: Éditions de l'École Polytechnique. pp. 1–83. ISBN 2-7302-1293-0. SIG 0222434. Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved" (PDF), Bollettino dell'American Mathematical Society, 41 (1): 43–57, doi:10.1090/S0273-0979-03-00993-5, SIG 2015449 Mihăilescu, Preda (2004), "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", J. Reine Angew. Matematica., 2004 (572): 167–195, doi:10.1515/crll.2004.048, SIG 2076124 Mihăilescu, Preda (2005), "Reflection, Bernoulli numbers and the proof of Catalan's conjecture" (PDF), European Congress of Mathematics, Zurich: Eur. Matematica. soc.: 325–340, SIG 2185753 Ribenboim, Paulo (1994), Catalan's Conjecture, Boston, MA: Stampa accademica, Inc., ISBN 0-12-587170-8, SIG 1259738 Predates Mihăilescu's proof. Tijdeman, Roberto (1976), "On the equation of Catalan" (PDF), Acta Arith., 29 (2): 197–209, doi:10.4064/aa-29-2-197-209, SIG 0404137 Weissstein esterno sinistro, Eric W. "Catalan's conjecture". Math World. Ivars Peterson's MathTrek On difference of perfect powers Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture hide Authority control National libraries France (data)Germany Other SUDOC (Francia) 1 Categorie: ConjecturesConjectures that have been provedDiophantine equationsTheorems in number theory

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