In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equation \[ a^m + b^n = c^k\quad \] (Eq.1) has only finitely many solutions (a,b,c,m,n,k); here a, b, c are positive coprime integers and m, n, k are positive integers satisfying \[ \frac{1}{m}+\frac{1}{n}+\frac{1}{k}<1. \] (Eq.2) As of 2008, the following solutions to Eq.1 are known:[1] \[ 1^m+2^3=3^2\; \] The first of these (1m+23=32) is the only solution where one of a, b or c is 1; this is the Catalan conjecture, proven in 2002 by Preda Mihăilescu. Technically, this case leads infinitely many solutions of Eq.1 (since we can pick any m for m>6), but for the purposes of the statement of the FermatCatalan conjecture we count all these solutions as one. It is known by Faltings' theorem that for any fixed choice of positive integers m, n and k satisfying Eq.2, only finitely many coprime triples (a, b, c) solving Eq.1 exist, but of course the full Fermat–Catalan conjecture is a much stronger statement. The abc conjecture implies the Fermat–Catalan conjecture.[1] ^ a b Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; BarrowGreen, June; Leader, Imre, The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 9780691118802. Retrieved from "http://en.wikipedia.org/"

