The conjecture states that the equation has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The inequality on m, n, and k is a necessary part of the conjecture.
As of 2015 the following ten solutions to equation (1) which meet the criteria of equation (2) are known:[1] The first of these (1m + 23 = 32) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu.
While this case leads to infinitely many solutions of (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (am, bn, ck).
It is known by the Darmon–Granville theorem, which uses Faltings's theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (a, b, c) solving (1) exist.[2][3]: p.
64 However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents m, n and k to vary.