4. If $a<b<c<d<e$ are consecutive positive integers, $b+c+d$ is a perfect square, and $a+b+c+d+e$ is a perfect cube, what is the minimum value of $c$?
$$
\begin{aligned}
4 . & b+c+d=3 c \\
& a+b+c+a+e=5 c
\end{aligned}
$$

and $b+c+d=m^{2}, a+b+c+d+e=n^{3}$
$$
\begin{aligned}
\therefore \quad 3 c & =m^{2}, \\
5 c & =n^{3} .
\end{aligned}
$$
$$
\begin{array}{l}
\therefore 3^{3} \mid c_{0} \text { and } 5\left|n_{0} \quad \therefore 5^{2}\right| c, \quad \therefore 25 \cdot 27 \mid c_{0} \\
\therefore \quad c=25 \cdot 27=675 .
\end{array}
$$