Suppose that $(k,l,m) \in \mathbb{N\_0}^3$.


If $(k,l,m)=(0,0,0)$ then for $n=1,2$ there is an infinite number of solutions and, by the theorem of of Wiles there are no solutions when $n \geq 3$.


Is it known can there be an infinite number of solutions of $a^{n+k}+b^{n+l}=c^{n+m}$ if $k,l,m$ are not all equal and $n \geq 3$?


What is known of when there is a finite non-zero number of solutions?


Can someone summarize all that is known about this generalized Fermat problem?


We see that a mapping $m: (k,l,m) \to a^{n+k}+b^{n+l}=c^{n+m}$ is a mapping that sends some element of $\mathbb{N\_0}^3$ to a family of equations over the integers and because every such equation has its own set of solutions for every $n \in \mathbb N$ we can define some mapping $s$ that sends some element of $\mathbb{N\_0}^3$ to a set of all solutions.


A question can then be phrased of when is $s(k,l,m)$ an empty set and when it is not an empty set.


We could probably somehow lower "dimensionality" of this problem by choosing some $(k,l,m)$ and fixing either $a$ or $b$ or $c$ but I am not sure how general results could be obtained in such a way.


Also, I do not know are techniques of Wiles suited for this more general problem, and, if they are not, I do not know why they are not?


It would be nice if someone could explain in non-strict technical sense all that is known about this generalized Fermat problem.


If this is off-topic and written in non-professional way then please vote to close this question of mine.



What is known about equation $a^{n+k}+b^{n+l}=c^{n+m}$ and its set of solutions?
[Darmon and Granville](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms/27.6.513) proved, using [Faltings' Theorem](https://en.wikipedia.org/wiki/Faltings%27s_theorem), that your equation has finitely many primitive integer solutions for any fixed exponents which are at least $3$. In fact their result is more general. For several concrete exponents beyond Fermat's Last Theorem, the full set of primitive integer solutions is also known, see e.g. the papers of [Siksek-Stoll](https://arxiv.org/abs/1309.4421) and [Anni-Siksek](https://arxiv.org/abs/1506.02860).


You can find more information in the [Wikipedia article on Beal's conjecture](https://en.wikipedia.org/wiki/Beal_conjecture) and the references therein. See also [this survey](https://homepages.warwick.ac.uk/~maseap/papers/bealconj.pdf), especially Section 4.5.


