To find least possible value of a no. in A.P. Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a + b + c + d + e$ is the cube of
an integer and $b + c + d$ is the square of an integer. The least possible value of the number of digits of $c$ is

Let the common difference be $\alpha$.
$$a+b+c+d+e = 5c$$
$$b+c+d=3c$$
Hence $5c$ is a cube and $3c$ is a square.
Hence $c=5^2k^3$ and $c=3l^2$.
$k$ must be divisible by $3$ and $l$ must be divisible by $5$.
$c=5^2.3^3.k_1^3$ and $c=5^2.3.l_1^2$
$$c=5^2.3.3^2k_1^3$$
Picking $k_1=1$ satisfies the condition.
