Generalize Fermat's Last Theorem

Question

Let $\sum_{j=1}^k a_j^n=z^n$. All $a_j,z$ positive integers. $k,n\gt 2$. For a given $n$, for what values of $k$ are there any solutions, and are there only finitely many? For $n=3$, there are solutions for $k=3$. Has this question been studied in detail?

Answer 1

This question closely resembles Euler's sum of powers conjecture. Based on the amount of literature on that problem and others inspired by it, I would say the question has been studied in detail. Whether that amount of detail is enough to satisfy your curiosity is another question, but perhaps that keyword gets you started.

Euler's conjecture says, in your notation, that solutions exists only if $k\geq n$, but that is known to be false in general. The unfortunate reality is that the full answer is not known to Euler's conjecture, and therefore your question as its generalization is also in the unknown realm. Some things are known, but there is no complete picture.