Available in electronic format
Available in print format		 Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues: 1891 - Present
Previous Article | Next Article
     

Catalan's Conjecture: Another old Diophantine problem solved

Author(s): Tauno Metsänkylä
Journal: Bull. Amer. Math. Soc. 41 (2004), 43-57.
MSC (2000): Primary 11D41, 00-02; Secondary 11R18
Posted: September 5, 2003
Retrieve article in: PDF DVI PostScript

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:

[ ]

  • For a more complete bibliography see the references given in [RIB], [MI2], [BIL] and [M2].



[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: 10.1090/S0273-0979-03-00993-5
PII: S 0273-0979(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
Posted: September 5, 2003
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google