<s>[INST] Show that $3^n+4^n+\dots+(n+2)^n=(n+3)^n$ has no answers for $n\ge 6$. Considering
$$3^n+4^n+\dots+(n+2)^n=(n+3)^n$$
Clearly $n=2$ and $n=3$ are solutions of this equation and this equality does not hold for $n=4$ and $n=5$.

How can I show this equation has no solutions for $n>5$.

Thanks.
 [/INST] A direct proof for $n\ge7$:
One can very easily show by induction $\sum_{k=1}^{n-1} k^n<n^n$. This implies \begin{align} \sum_{k=3}^{n+2}\left(\frac{k}{n}\right)^n &=1+\left(1+\frac1n\right)^n+\left(1+\frac2n\right)^n+\sum_{k=3}^{n-1} \left(\frac{k}{n}\right)^n \\ &< 2+e+e^2<\left(\frac{10}{7}\right)^7\le\left(1+\frac3n\right)^n.\end{align}
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