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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:
3^{2}  2^{3} = 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^{.}10^{16} and the
smaller is at least 10^{7} [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^{q1} 1 (mod q^{2})
q^{p1} 1 (mod p^{2})
That is p is a Wieferich prime base q, and q
is a Wieferich prime base p. (Others had shown this for certain subsets ofthe primes.)
FermatCatalan equation
Solutions to Catalan's conjecture and Fermat's Last Theorem
are special cases of the FermatCatalan 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 FermatCatalan conjecture is that there are only finitely many
solutions to this system. These solutions include:
1^{p} + 2^{3} = 3^{2} (p > 2)
2^{5} + 7^{2} = 3^{4}
13^{2} + 7^{3} = 2^{9}
2^{7} + 17^{3} = 71^{2}
3^{5} + 11^{4} = 122^{2}
33^{8} + 1549034^{2} = 15613^{3}
1414^{3} + 2213459^{2} = 65^{7}
9262^{3} + 15312283^{2} = 113^{7}
17^{7} + 76271^{3} = 21063928^{2}
43^{8} + 96222^{3} = 30042907^{2}
See Also: FermatsLastTheorem, BealsConjecture, WieferichPrime References:
 Catalan1844
 E. Catalan, "Note extraite d'une lettre adressée à l'édite," J. reine angew. Mathematik, 27 (1844) 192.
 CP2001 (p279381)
 R. Crandall and C. Pomerance, Prime numbers: a computational perspective, SpringerVerlag, 2001. New York, NY, pp. xvi+545, ISBN 0387947779. 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, SpringerVerlag, New York, NY, 1994. ISBN 0387942890. 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) 225231. 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 0125871708. MR 95a:11029
