Fermat–Catalan conjecture

In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation

${\displaystyle a^{m}+b^{n}=c^{k}\quad }$

1

has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (a^m, bⁿ, c^k) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying

${\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1.}$

2

The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = a^m + bⁿ), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. ${\displaystyle 7^{5}+393^{3}=7792^{2}}$.