3 Find all prime numbers $p$, such that $2^{p}+3^{p}$ is a $k$-th power of an integer (where $k \geqslant 2$).

Find all prime numbers $p$, such that $2^{p}+3^{p}$ is a $k$-th power of an integer (where $k \geqslant 2$).
3. $p=2,5$ do not meet the requirements. Let prime $p>2$ and $p \neq 5$. By the binomial theorem, it is easy to see that
$$\begin{aligned}
2^{p}+3^{p} & =2^{p}+(5-2)^{p}=5^{p}-\mathrm{C}_{p}^{1} 5^{p-1} \times 2+\cdots+5 \mathrm{C}_{p}^{p-1} 2^{p-1} \\
& =5^{2} u+5 p \times 2^{p-1}, u \text { is an integer. }
\end{aligned}$$

Thus $5 \|\left(2^{p}+3^{p}\right)$, hence $2^{p}+3^{p}$ cannot be a $k$-th power of an integer $(k>1)$.