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:
xp - yq = 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 yq 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:
pq-1 ≡ 1 (mod q2)
qp-1 ≡ 1 (mod p2)
Solutions to Catalan's conjecture and Fermat's Last Theorem are special cases of the Fermat-Catalan equation
xp + yq = zr
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:
1p + 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
- 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, New York, NY, 2001. 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.]
- 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
- P. Mihailescu, "A class number free criterion for Catalan's conjecture," J. Number Theory, 99:2 (2003) 225--231. MR 1 968 450
- 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.
- P. Ribenboim, Catalan's conjecture: are 8 and 9 the only consecutive powers?, Academic Press, Boston, MA, 1994. pp. xvi+364, ISBN 0-12-587170-8. MR 95a:11029