Catalan's problem

In 1844, the Belgian mathematician Eugène Charles Catalan conjectured that 8 and 9 are the only pair of consecutive powers. That is, the Catalan equation for primes p and q and positive integers x and y:

x^p - y^q = 1

has only the one solution:

3² - 2³ = 1

In 1976 R. Tijdeman took the first major step toward showing this by proving that for any solution y^q is less than e^e^e^e^730 (a huge number!) [Guy94]. Since then this bound has been reduced many times, and we now know that the larger of p and q is at most 7.78^.10¹⁶ and the smaller is at least 10⁷ [Mignotte2000].

On April 18, 2002, Preda Mihailescu published his completed proof [Mihailescu2003] which begins by showing that about solutions to this problem is that any solutions other than the pair (p,q) = (2,3) must satisfy both of:

p^q-1 ≡ 1 (mod q²)
q^p-1 ≡ 1 (mod p²)

That is, p is a Wieferich prime base q, and q is a Wieferich prime base p. (Others had shown this for certain subsets of the primes.)

Fermat-Catalan equation

Solutions to Catalan's conjecture and Fermat's Last Theorem are special cases of the Fermat-Catalan equation

x^p + y^q = z^r

Where x, y, z are positive, coprime integers and the exponents are all primes with

1/p + 1/q + 1/r ≤ 1.

The Fermat-Catalan conjecture is that there are only finitely many solutions to this system. These solutions include:

1^p + 2³ = 3² (p ≥ 2)
2⁵ + 7² = 3⁴
13² + 7³ = 2⁹
2⁷ + 17³ = 71²
3⁵ + 11⁴ = 122²
33⁸ + 1549034² = 15613³
1414³ + 2213459² = 65⁷
9262³ + 15312283² = 113⁷
17⁷ + 76271³ = 21063928²
43⁸ + 96222³ = 30042907²

References:

Catalan1844
E. Catalan, "Note extraite d'une lettre adressée à l'édite," J. reine angew. Mathematik, 27 (1844) 192.
CP2001 (p279-381)
R. Crandall and C. Pomerance, Prime numbers: a computational perspective, Springer-Verlag, 2001. New York, NY, pp. xvi+545, ISBN 0-387-94777-9. MR 2002a:11007 (Abstract available) [This is a valuable text written by true experts in two different areas: computational and theoretical respectively. There is now a second edition [CP2005].]
Guy94 (section D9)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, 1994. New York, NY, ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
Mignotte2000
M. Mignotte, Catalan's equation just before 2001. In "Number theory (Turku, 1999)," M. Jutila and T. Metsänkylä editors, de Gruyter, Berlin, 2001. MR 2002g:11034
Mihailescu2003
P. Mihailescu, "A class number free criterion for Catalan's conjecture," J. Number Theory, 99:2 (2003) 225--231. MR 1 968 450
Peterson2000
I. Peterson, "Prime proof zeros in on crucial numbers," Science News, 158 (December 2000) 357. Short note that Miailescu showed solutions to Catalan's are Wierferich double primes.
Ribenboim1994
P. Ribenboim, Catalan's conjecture: are 8 and 9 the only consecutive powers?, Academic Press, 1994. Boston, MA, pp. xvi+364, ISBN 0-12-587170-8. MR 95a:11029