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The Problem of Catalan pp 1–9Cite as

A Historical Account

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Abstract

This is a very brief overview of the history of Catalan’s problem.

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Notes

  1. 1.

    The journal was founded in the same year by the École Polytechnique teachers Gerono and Prouhet and was mainly addressed to the young candidates of the École Polytechnique and École Normale Supérieure.

  2. 2.

    Victor Amédée Lebesgue (1791–1875), not the much more famous Henri Léon Lebesgue.

  3. 3.

    Trygve Nagell (sometimes spelled as Nagel, 1895–1988).

  4. 4.

    Sigmund Selberg (1910–1994), the elder brother of Atle Selberg.

  5. 5.

    Kustaa Inkeri (1908–1997).

  6. 6.

    Recall that he was one of the founders of the Nouvelles Annales de Mathématiques.

  7. 7.

    A similar condition occurred in the work of Wieferich [138] on the Fermat equation; see Sect. 2.3.3.

References

  1. Baker, A.: Linear forms in the logarithms of algebraic numbers I–IV. Mathematika 13, 204–216 (1966); 14, 102–107, 220–224, (1967); 15 204–216 (1968)

    Google Scholar 

  2. Baker, A.: Bounds for solutions of hyperelliptic equations. Proc. Camb. Phil. Soc. 65, 439–444 (1969)

    Article  MATH  Google Scholar 

  3. Bennett M.A.: Review of “Catalan’s equation with a quadratic exponent”. Math. Rev. MR1816462 (2002a:11021)

    Google Scholar 

  4. Bilu, Yu.F.: Catalan’s conjecture (after Mihăilescu). Séminaire Bourbaki, Exposé 909, 55ème année (2002–2003)

    Google Scholar 

  5. Bilu, Yu.F.: Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu). J. Th. Nombres Bordeaux 17, 69–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bilu, Yu.F., Hanrot, G.: Solving superelliptic Diophantine equations by Baker’s method. Comp. Math. 112, 273–312 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bugeaud, Y.: Bounds for the solutions of superelliptic equations. Comp. Math. 107, 187–219 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bugeaud, Y., Hanrot, G.: Un nouveau critère pour l’équation de Catalan. Mathematika 47, 63–73 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carmichael, R.D.: Diophantine Analysis. Wiley, Chapman Hall, New York (1915)

    MATH  Google Scholar 

  10. Cassels, J.W.S.: On the equation a x − b y = 1. Am. J. Math. 75, 159–162 (1953)

    Google Scholar 

  11. Cassels, J.W.S.: On the equation a x − b y = 1 II. Proc. Camb. Philos. Soc. 56, 97–103 (1960); Corrigendum: Ibid. 57, 187 (1961)

    Google Scholar 

  12. Catalan, E.: Problème 48. Nouv. Ann. Math. 1, 520 (1842)

    Google Scholar 

  13. Catalan, E.: Note extraite d’une lettre adressée à l’éditeur. J. Reine Angew. Math. 27, 192 (1844)

    Article  MATH  Google Scholar 

  14. Dickson, L.E.: History of the Theory of Numbers II: Diophantine Analysis. G. E. Stechert & Co., New York (1934)

    Google Scholar 

  15. Evertse, J.-H.: Upper Bounds for the Numbers of Solutions of Diophantine Equations. Mathematical Centre Tracts, vol. 168. Mathematisch Centrum, Amsterdam (1983)

    Google Scholar 

  16. Euler, L.: Theorematum quorundam arithmeticorum demonstrationes. Comm. Acad. Sci. Petrop. 10, 125–146 (1738); Opera Omnia Ser. I, Vol. I, Commentationes Arithmeticae I, 38–58. Teubner, Basel (1915)

    Google Scholar 

  17. Gerono, G.C.: Sur la résolution en nombres entiers et positifs de l’équation x m = y n + 1. Nouv. Ann. Math. 16, 394–398 (1857)

    Google Scholar 

  18. Gerono, G.C.: Sur la résolution en nombres entiers et positifs de l’équation x m = y n + 1. Nouv. Ann. Math. (2) 9, 469–471 (1870)

    Google Scholar 

  19. Gerono, G.C.: Sur la résolution en nombres entiers et positifs de l’équation x m = y n + 1. Nouv. Ann. Math. (2) 10, 204–206 (1871)

    Google Scholar 

  20. Glass, A.M.W., Meronk, D.B., Okada, T., Steiner, R.P.: A small contribution to Catalan’s equation. J. Number Theory 47, 131–137 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gloden, A.: Sur un problème de Catalan. Mathesis 61, 302–303 (1952)

    MathSciNet  Google Scholar 

  22. Gloden, A.: Histoire du ‘problème de Catalan’. In: Actes du VIIe Congrès International d’histoire des Sciences (Jérusalem 4–12 Août 1953), pp. 316–319. Académie Internationale d’Histoire des Sciences, Paris (1953)

    Google Scholar 

  23. Hampel, R.: On the solution in natural numbers of the equation x m − y n = 1. Ann. Polon. Math. 3, 1–4 (1956)

    Google Scholar 

  24. Hyyrö, S.: Catalan’s problem (Finnish). Arkhimedes 1, 53–54 (1963)

    Google Scholar 

  25. Hyyrö, S.: Über das Catalan’sche problem. Ann. Univ. Turku Ser. AI 79, 3–10 (1964)

    Google Scholar 

  26. Inkeri, K: On Catalan’s problem. Acta Arith. 9, 285–290 (1964)

    MathSciNet  MATH  Google Scholar 

  27. Inkeri, K: On Catalan’s conjecture. J. Number Theory 34, 142–152 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Inkeri, K., Hyyrö, S.: On the congruence \(3^{p-1} \equiv 1\,(\ \mathrm{mod}\,p^{2})\) and the Diophantine equation x 2 − 1 = y p. Ann. Univ. Turku. Ser. A I No. 50, 1–4 (1961)

    Google Scholar 

  29. Ko, C.: On the diophantine equation x 2 = y n + 1, xy ≠ 0. Sci. Sinica 14, 457–460 (1965)

    Google Scholar 

  30. Langevin, M.: Quelques applications de nouveaux résultats de Van der Poorten. In: Séminaire Delange-Pisot-Poitou, 17e année (1975/1976), Théorie des nombres: Fasc. 2, Exp. No. G12, 11 pp. Secrétariat Math., Paris (1977)

    Google Scholar 

  31. Lebesgue, V.A.: Sur l’impossibilité en nombres entiers de l’équation x m = y 2 + 1. Nouv. Ann. Math. 9, 178–181 (1850)

    Google Scholar 

  32. LeVeque, W.J.: On the equation a x − b y = 1. Am. J. Math. 74, 325–331 (1952)

    Google Scholar 

  33. LeVeque, W.J.: Topics in Number Theory. Vol. II. Addison-Wesley Publishing Co. Inc., Reading (1956)

    Google Scholar 

  34. Lionnet, E.: Question 884. Nouv. Ann. Math. (2) 7, 240 (1868)

    Google Scholar 

  35. Ma̧kowski, A.: Three consecutive integers cannot be powers. Colloq. Math. 9, 297 (1962)

    Google Scholar 

  36. Meyl, A.J.F.: Question 1196. Nouv. Ann. Math. (2) 15, 545–547 (1876)

    Google Scholar 

  37. Mignotte, M.: Sur l’équation de Catalan. C. R. Acad. Sci. Paris 314, 165–168 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Mignotte, M.: Un critère élémentaire pour l’équation de Catalan. C. R. Math. Rep. Acad. Sci. Can. 15, 199–200 (1993)

    MathSciNet  MATH  Google Scholar 

  39. Mignotte, M.: A criterion on Catalan’s equation. J. Number Theory 52, 280–283 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mignotte, M.: Catalan’s equation just before 2000. In: Number Theory (Turku, 1999), pp. 247–254. de Gruyter, Berlin (2001)

    Google Scholar 

  41. Mignotte, M., Roy, Y.: Catalan’s equation has no new solution with either exponent less than 10651. Exp. Math. 4, 259–268 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mignotte, M., Roy, Y.: Minorations pour l’équation de Catalan. C. R. Acad. Sci. Paris 324, 377–380 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mihăilescu, P.: A class number free criterion for Catalan’s conjecture. J. Number Theory 99, 225–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mihăilescu, P.: Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)

    MathSciNet  MATH  Google Scholar 

  45. Mihăilescu, P.: On the class groups of cyclotomic extensions in the presence of a solution to Catalan’s equation. J. Number Theory 118, 123–144 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Moret-Blanc, M.: Question 1175. Nouv. Ann. Math. (2) 15, 44–46 (1876)

    Google Scholar 

  47. Nagell, T.: Problem 39. Norsk Mat. Tidsskrift 1, 164 (1919)

    Google Scholar 

  48. Nagell, T.: Des équations indéterminées x 2 + x + 1 = y n et x 2 + x + 1 = 3y n. Norsk. Mat. Forenings Skrifter (1) 2, 12–14 (1921)

    Google Scholar 

  49. Nagell, T.: Sur l’impossibilité de l’équation indéterminée z p + 1 = y 2. Norsk Mat. Forenings Skrifter (1) 4, 14 pp (1921)

    Google Scholar 

  50. Nagell, T.: Sur une équation diophantienne à deux indéterminées. Norsk. Vid. Selsk. Forenings Trondheim 7, 137–139 (1934)

    Google Scholar 

  51. Obláth, R.: On the numbers x 2 − 1 (Hungarian). Mat. Fiz. Lapok 47, 58–77 (1940)

    MathSciNet  Google Scholar 

  52. Obláth, R.: On impossible Diophantine equations of the form x m + 1 = y n (Spanish). Revista Mat. Hisp.-Am. 1, 122–140 (1941)

    Google Scholar 

  53. Obláth, R.: Über die Gleichung x m + 1 = y n. Ann. Polon. Math. 1, 73–76 (1954)

    Google Scholar 

  54. Poirier, H.: Il a démontré la conjecture de Catalan. Science et Vie 1020, 70–73 (2002)

    Google Scholar 

  55. Ribenboim, P.: Catalan’s Conjecture. Academic, Boston (1994)

    MATH  Google Scholar 

  56. Rotkiewicz, A.: Sur l’équation x z − y t = a t, où \(\vert x - y\vert = a\). Ann. Polon. Math. 2, 7–8 (1956)

    Google Scholar 

  57. Schinzel, A.: Sur l’équation x z − y t = 1, où \(\vert x - y\vert = 1\). Ann. Polon. Math. 3, 5–6 (1956)

    Google Scholar 

  58. Schinzel, A., Tijdeman, R.: On the equation y m = P(x). Acta Arith. 31, 199–204 (1976)

    Google Scholar 

  59. Schoof, R.: Catalan’s Conjecture. Universitext. Springer, London (2008)

    MATH  Google Scholar 

  60. Schwarz, W.: A note on Catalan’s equation. Acta Arith. 72, 277–279 (1995)

    MathSciNet  MATH  Google Scholar 

  61. Selberg, S.: Solution of problem 39 (Norvegian). Norsk Mat. Tidsskrift 14, 79–80 (1932)

    MathSciNet  Google Scholar 

  62. Siegel, C.L. (under the pseudonym X): The integer solutions of the equation \(y^{2}\, =\, ax^{n} + bx^{n-1} + \cdots + k\) (Extract from a letter to Prof. L. J. Mordell). J. Lond. Math. Soc. 1, 66–68 (1926)

    Google Scholar 

  63. Sierpiński, W.: On some unsolved problems of arithmetics. Scripta Math. 25, 125–136 (1960)

    MathSciNet  MATH  Google Scholar 

  64. Tijdeman, R.: On the equation of Catalan. Acta Arith. 29, 197–209 (1976)

    MathSciNet  Google Scholar 

  65. Wieferich, A.: Zum letzten Fermat’schen theorem. J. Reine Angew. Math. 136, 293–302 (1909)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of Bordeaux, University of Bordeaux and CNRS, Talence, France

    Yuri F. Bilu

  2. IRMA, Mathematical Institute, University of Strasbourg and CNRS, Strasbourg, France

    Yann Bugeaud & Maurice Mignotte

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  1. Yuri F. Bilu
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  2. Yann Bugeaud
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  3. Maurice Mignotte
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Bilu, Y.F., Bugeaud, Y., Mignotte, M. (2014). A Historical Account. In: The Problem of Catalan. Springer, Cham. https://doi.org/10.1007/978-3-319-10094-4_1

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