Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)

Yuri F. Bilu[1]

  • [1] A2X, Université Bordeaux 1 351 cours de la Libération 33405 Talence France

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 69-85
  • ISSN: 1246-7405

Abstract

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This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.

How to cite

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Bilu, Yuri F.. "Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 69-85. <http://eudml.org/doc/249443>.

@article{Bilu2005,
abstract = {This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.},
affiliation = {A2X, Université Bordeaux 1 351 cours de la Libération 33405 Talence France},
author = {Bilu, Yuri F.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {69-85},
publisher = {Université Bordeaux 1},
title = {Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)},
url = {http://eudml.org/doc/249443},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Bilu, Yuri F.
TI - Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 69
EP - 85
AB - This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.
LA - eng
UR - http://eudml.org/doc/249443
ER -

References

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  1. Yu.F. Bilu, Catalan’s conjecture (after Mihăilescu). Séminaire Bourbaki, Exposé 909, 55ème année (2002-2003); Astérisque 294 (2004), 1–26. Zbl1094.11014MR2111637
  2. Y. Bugeaud, G. Hanrot, Un nouveau critère pour l’équation de Catalan. Mathematika 47 (2000), 63–73. Zbl1008.11011MR1924488
  3. J. W. S. Cassels, On the equation a x - b y = 1 , II. Proc. Cambridge Philos. Society 56 (1960), 97–103. Zbl0094.25702MR114791
  4. E. Catalan, Note extraite d’une lettre adressée à l’éditeur. J. reine angew. Math. 27 (1844), 192. 
  5. S. Hyyrö, Über das Catalansche Problem. Ann. Univ. Turku Ser. AI 79 (1964), 3–10. Zbl0127.01904MR179127
  6. P. Kirschenhofer, A. Pethő, R.F. Tichy, On analytical and Diophantine properties of a family of counting polynomials. Acta Sci. Math. (Szeged), 65 (1999), no. 1-2, 47–59. Zbl0983.11013MR1702180
  7. Ko Chao, On the diophantine equation x 2 = y n + 1 , x y ≠ 0 . Sci. Sinica 14 (1965), 457–460. Zbl0163.04004MR183684
  8. E. KummerCollected papers. Springer, 1975. Zbl0327.01019MR465761
  9. V.A. Lebesgue, Sur l’impossibilité en nombres entiers de l’équation x m = y 2 + 1 . Nouv. Ann. Math. 9 (1850), 178–181. 
  10. M. Laurent, M. Mignotte, Yu. Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation. J. Number Theory 55 (1995), 285–321. Zbl0843.11036MR1366574
  11. M. Mignotte, Catalan’s equation just before 2000. Number theory (Turku, 1999), de Gruyter, Berlin, 2001, pp. 247–254. Zbl1065.11019MR1822013
  12. M. Mignotte, Y. Roy, Catalan’s equation has no new solutions with either exponent less than 10651 . Experimental Math. 4 (1995), 259–268. Zbl0857.11012MR1387692
  13. M. Mignotte, Y. Roy, Minorations pour l’équation de Catalan. C. R. Acad. Sci. Paris 324 (1997), 377–380. Zbl0887.11018MR1440951
  14. P. Mihăilescu, A class number free criterion for Catalan’s conjecture. J. Number Theory 99 (2003), 225–231. Zbl1049.11036MR1968450
  15. P. Mihăilescu, Primary cyclotomic units and a proof of Catalan’s conjecture. J. reine angew. Math., to appear. Zbl1067.11017MR2076124
  16. P. Mihăilescu, On the class groups of cyclotomic extensions in the presence of a solution to Catalan’s equation. A manuscript. Zbl1104.11049
  17. T. Nagell, Des équations indéterminées x 2 + x + 1 = y n and x 2 + x + 1 = 3 y n . Norsk Matem. Forenings Skrifter I, 2 (1921), 14 pp. (See also: Collected papers of Trygve Nagell, ed. P. Ribenboim, Queens Papers in Pure and Applied Mathematics 121, Kingston, 2002; Vol.1, pp. 79–94.) 
  18. R. Tijdeman, On the equation of Catalan. Acta Arith. 29 (1976), 197–209. Zbl0286.10013MR404137
  19. L. Washington, Introduction to Cyclotomic Fields. Second edition, Graduate Texts in Math. 83, Springer, New York, 1997. Zbl0966.11047MR1421575

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