# A Generalised Diophantine Conjecture

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?!

$$95800^4 + 217519^4 + 414560^4 = 422481^4$$
$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$