19. 1. 21 ** If $a<b<c<d<e$ are consecutive positive integers, $b+c+d$ is a square number, $a+b+c+d+e$ is a cube number, what is the minimum value of $c$?
Parse the cubic number $a+b+c+d+e=5^{3} \times k^{3}, k \in \mathbf{N}$ (since the sum of 5 consecutive natural numbers is a multiple of 5). The square number $b+c+d=\frac{3}{5}(a+b+c+d+e)=5^{2} \times 3 \times k^{3}$. Therefore, the minimum value of $k$ is $3$, and the minimum value of $c=5^{2} \times k^{3}$ is $5^{2} \times 3^{3}=675$.