Fermat–Catalan conjecture

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]

\[ 1^m+2^3=3^2\; \]
\[ 2^5+7^2=3^4\; \]
\[ 13^2+7^3=2^9\; \]
\[ 2^7+17^3=71^2\; \]
\[ 3^5+11^4=122^2\; \]
\[ 33^8+1549034^2=15613^3\; \]
\[ 1414^3+2213459^2=65^7\; \]
\[ 9262^3+15312283^2=113^7\; \]
\[ 17^7+76271^3=21063928^2\; \]
\[ 43^8+96222^3=30042907^2\; \]

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 Fermat-Catalan 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]
References

^ a b Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre, The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 9780691118802.