Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Catalan's Conjecture: Another old Diophantine problem solved


Author: Tauno Metsänkylä
Journal: Bull. Amer. Math. Soc. 41 (2004), 43-57
MSC (2000): Primary 11D41, 00-02; Secondary 11R18
DOI: https://doi.org/10.1090/S0273-0979-03-00993-5
Published electronically: September 5, 2003
MathSciNet review: 2015449
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Catalan's Conjecture predicts that 8 and 9 are the only consecutive perfect powers among positive integers. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihailescu. A deep theorem about cyclotomic fields plays a crucial role in his proof.

Like Fermat's problem, this problem has a rich history with some surprising turns. The present article surveys the main lines of this history and outlines Mihailescu's brilliant proof.


References [Enhancements On Off] (What's this?)

  • [BIL] Y.F. Bilu, Catalan's conjecture $[$after Mihailescu$]$, Sém. Bourbaki, 55ème année, n$^{o}$ 909 (2002/03), 24 pp.
  • [BH] Y. Bugeaud, G. Hanrot, Un nouveau critère pour l'équation de Catalan, Mathematika 47 (2000), 63-73. MR 2003h:11039
  • [CAS] J.W.S. Cassels, On the equation $a^{x}-b^{y}=1$,II, Proc. Cambridge Philos. Soc. 56 (1960), 97-103. MR 22:5610
  • [CAT] E. Catalan, Note extraite d'une lettre adressée à l'éditeur, J. Reine Angew. Math. 27 (1844), 192.
  • [CHE] E.Z. Chein, A note on the equation $x^{2}=y^{q}+1$, Proc. Amer. Math. Soc. 56 (1976), 83-84. MR 53:7937
  • [COH] P.M. Cohn, Algebra, Vol. 2, John Wiley & Sons, Chichester - New York, 1977. MR 58:26625
  • [DIL] K. Dilcher, Fermat numbers, Wieferich and Wilson primes: computations and generalizations, Public-key cryptography and computational number theory (Warsaw, 2000), de Gruyter, Berlin, 2001, pp. 29-48. MR 2002j:11004
  • [GRE] R. Greenberg, On $p$-adic $L$-functions and cyclotomic fields, II, Nagoya Math. J. 67 (1977), 139-158. MR 56:2964
  • [HY1] S. Hyyrö, Über das Catalansche Problem, Ann. Univ. Turku, Ser. A I no. 79 (1964), 8 pp. MR 31:3378
  • [HY2] S. Hyyrö, Über die Gleichung $ax^{n}-by^{n}=z$und das Catalansche Problem, Ann. Acad. Sci. Fenn., Ser. A I no. 355 (1964), 50 pp. MR 34:5750
  • [IN1] K. Inkeri, On Catalan's problem, Acta Arith. 9 (1964), 285-290. MR 29:5780
  • [IN2] K. Inkeri, On Catalan's conjecture, J. Number Theory 34 (1990), 142-152. MR 91e:11030
  • [KER] W. Keller, J. Richstein, Solutions of the congruence $a^{p-1}\equiv 1 \pmod {p^{r}}$, Math. Comput. (to appear).
  • [KNR] J. Knauer, J. Richstein, The continuing search for Wieferich primes, preprint (2003).
  • [KO] Chao Ko [Ko Chao], On the Diophantine equation $x^{2}=y^{n}+1,\, xy\neq 0$, Sci. Sinica (Notes) 14 (1964), 457-460. MR 32:1164
  • [LEB] V.A. Lebesgue, Sur l'impossibilité en nombres entiers de l'équation $x^{m}=y^{2}+1$, Nouv. Ann. Math. 9 (1850), 178-181.
  • [LET] G. Lettl, A note on Thaine's circular units, J. Number Theory 35 (1990), 224-226. MR 91h:11118
  • [MW] B. Mazur, A. Wiles, Class fields of abelian extensions of $\mathbb{Q} $, Invent. Math. 76 (1984), 179-330. MR 85m:11069
  • [MI1] M. Mignotte, A criterion on Catalan's equation, J. Number Theory 52 (1995), 280-283. MR 96b:11042
  • [MI2] M. Mignotte, Catalan's equation just before $2000$, Number Theory (Turku, 1999), de Gruyter, Berlin, 2001, pp. 247-254. MR 2002g:11034
  • [M1] P. Mihailescu, A class number free criterion for Catalan's conjecture, J. Number Theory 99 (2003), 225-231.
  • [M2] P. Mihailescu, Primary cyclotomic units and a proof of Catalan's conjecture, preprint (September 2, 2002), submitted.
  • [NAG] T. Nagell, Sur l'impossibilité de l'équation indéterminée $z^{p}+1=y^{2}$, Norsk Mat. Forenings Skrifter, Ser. I no. 4 (1921), 10 pp.
  • [NAT] M.B. Nathanson, Catalan's equation in $K(t)$, Amer. Math. Monthly 81 (1974), 371-373. MR 49:218
  • [RIB] P. Ribenboim, Catalan's Conjecture, Academic Press, Boston, 1994. MR 95a:11029
  • [SCH] W. Schwarz, A note on Catalan's equation, Acta Arith. 72 (1995), 277-279. MR 96f:11048
  • [STE] R. Steiner, Class number bounds and Catalan's equation, Math. Comput. 67 (1998), 1317-1322. MR 98j:11021
  • [THA] F. Thaine, On the ideal class groups of real abelian number fields, Ann. of Math. 128 (1988), 1-18. MR 89m:11099
  • [TIJ] R. Tijdeman, On the equation of Catalan, Acta Arith. 29 (1976), 197-209. MR 53:7941
  • [WAS] L.C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer-Verlag, New York-Berlin-Heidelberg, 1997. MR 97h:11130

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11D41, 00-02, 11R18

Retrieve articles in all journals with MSC (2000): 11D41, 00-02, 11R18


Additional Information

Tauno Metsänkylä
Affiliation: Department of Mathematics, University of Turku, FIN-20014 Turku, Finland
Email: taumets@utu.fi

DOI: https://doi.org/10.1090/S0273-0979-03-00993-5
Keywords: Catalan's Conjecture, Diophantine equations of higher degree, cyclotomic fields, research exposition
Received by editor(s): March 5, 2003
Received by editor(s) in revised form: July 14, 2003
Published electronically: September 5, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society