Catalan’s Conjecture: Another old Diophantine problem solved

Metsänkylä, Tauno

doi:10.1090/S0273-0979-03-00993-5

Catalan’s Conjecture: Another old Diophantine problem solved
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Bull. Amer. Math. Soc. 41 (2004), 43-57 Request permission

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 Mihăilescu. 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 Mihăilescu’s brilliant proof.

References

Yann Bugeaud and Guillaume Hanrot, Un nouveau critère pour l’équation de Catalan, Mathematika 47 (2000), no. 1-2, 63–73 (2002) (French, with English and French summaries). MR 1924488, DOI 10.1112/S0025579300015722
J. W. S. Cassels, On the equation $a^{x}-b^{y}=1$. II, Proc. Cambridge Philos. Soc. 56 (1960), 97–103. MR 114791, DOI 10.1017/s0305004100034332

Note extraite d’une lettre adressée à l’éditeur

27

E. Z. Chein, A note on the equation $x^{2}=y^{q}+1$, Proc. Amer. Math. Soc. 56 (1976), 83–84. MR 404133, DOI 10.1090/S0002-9939-1976-0404133-1
Paul Moritz Cohn, Algebra. Vol. 2, John Wiley & Sons, London-New York-Sydney, 1977. With errata to Vol. I. MR 0530404
Karl 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 1881625
Ralph Greenberg, On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J. 67 (1977), 139–158. MR 444614, DOI 10.1017/S0027763000022583
Seppo Hyyrö, Über das Catalansche Problem, Ann. Univ. Turku. Ser. A I 79 (1964), 10 (German). MR 179127
Seppo Hyyrö, Über die Gleichung $ax^{n}-by^{n}=z$ und das Catalansche Problem, Ann. Acad. Sci. Fenn. Ser. A I No. 355 (1964), 50 (German). MR 0205925
K. Inkeri, On Catalan’s problem, Acta Arith. 9 (1964), 285–290. MR 168518, DOI 10.4064/aa-9-3-285-290
K. Inkeri, On Catalan’s conjecture, J. Number Theory 34 (1990), no. 2, 142–152. MR 1042488, DOI 10.1016/0022-314X(90)90145-H

Solutions of the congruence $a^{p-1}\equiv 1 \pmod {p^{r}}$

The continuing search for Wieferich primes

Chao Ko, On the Diophantine equation $x^{2}=y^{n}+1,\,xy\not =0$, Sci. Sinica 14 (1965), 457–460. MR 183684

Sur l’impossibilité en nombres entiers de l’équation $x^{m}=y^{2}+1$

9

Günter Lettl, A note on Thaine’s circular units, J. Number Theory 35 (1990), no. 2, 224–226. MR 1057325, DOI 10.1016/0022-314X(90)90115-8
B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
Maurice Mignotte, A criterion on Catalan’s equation, J. Number Theory 52 (1995), no. 2, 280–283. MR 1336750, DOI 10.1006/jnth.1995.1070
Maurice Mignotte, Catalan’s equation just before 2000, Number theory (Turku, 1999) de Gruyter, Berlin, 2001, pp. 247–254. MR 1822013

A class number free criterion for Catalan’s conjecture

99

Primary cyclotomic units and a proof of Catalan’s conjecture

Sur l’impossibilité de l’équation indéterminée $z^{p}+1=y^{2}$

Melvyn B. Nathanson, Catalan’s equation in $K(t)$, Amer. Math. Monthly 81 (1974), 371–373. MR 335436, DOI 10.2307/2319001
Paulo Ribenboim, Catalan’s conjecture, Academic Press, Inc., Boston, MA, 1994. Are $8$ and $9$ the only consecutive powers?. MR 1259738
Wolfgang Schwarz, A note on Catalan’s equation, Acta Arith. 72 (1995), no. 3, 277–279. MR 1347490, DOI 10.4064/aa-72-3-277-279
Ray Steiner, Class number bounds and Catalan’s equation, Math. Comp. 67 (1998), no. 223, 1317–1322. MR 1468945, DOI 10.1090/S0025-5718-98-00966-1
Francisco Thaine, On the ideal class groups of real abelian number fields, Ann. of Math. (2) 128 (1988), no. 1, 1–18. MR 951505, DOI 10.2307/1971460
R. Tijdeman, On the equation of Catalan, Acta Arith. 29 (1976), no. 2, 197–209. MR 404137, DOI 10.4064/aa-29-2-197-209
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7