I submitted without proof (and verification), the following conjecture:
The Diophantine equation
$$\sum_{k=1}^{L} (x_k)^n = y^n$$
has integer solutions only for $n\le L$.
Fermat’s Last Theorem is the special case of $L=2, n>2$. Another special case $n=L=2$ is that of the Pythagorean Triples.
However, presented with two counterexamples (one more than necessary!) I withdraw the aforesaid conjecture, while noting that it is really interesting that it does apply somewhat specifically to $L=2$ (Fermat's Case!)
So A better question to pose is : Seeing as how it does not hold for $L=3$ and $L=4$ , is $L=2$ the only known case or are there possibly other values of $L$ for which it does hold?!
$$ 95800^4 + 217519^4 + 414560^4 = 422481^4 $$
$$ 27^5 + 84^5 + 110^5 + 133^5 = 144^5 $$