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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?!

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$$ 95800^4 + 217519^4 + 414560^4 = 422481^4 $$

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

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