Integer solutions to equations of the form $a^n+b^n+\cdots=c^n$

Question

I shall refer to the number of terms on the left side of the equation as $m$. Suppose that all numbers in the equation are positive integers. I am wondering if anything is known about for which values of $m$ and $n$ this equation has infinitely many or no solutions. I know that the case for $m=2$ is solved (this reduces to Fermat's Last Theorem when $m=2$) and that $m>2$ are trivial, but is anything known for higher $m$? If so, is there a type of "generalized Fermat's Last Theorem" to encompass these cases with $m>2$?

Answer 1

Euler himself conjectured that for all solutions, $m$ must be at least $n$.

Unfortunately this was not true, because

$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$

A similar conjecture was made in 1967, and has not been proven or disproven yet as of now.