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

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

 

 

 

 

()

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

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

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