The sum of three consecutive positive integers is a perfect cube. find the least possible value of the smallest number, assuming that the middle number ends in at least 5 zeros.

Let us say that,

x = middle number

x - 1 = previous number

x + 1 = next number

The sum of the three is:

x - 1 + x + x + 1 = 3 x

If 3x is a perfect cube then x must be (3 * 3) to complete three 3’s.

x = 9

x - 1 = 8

x + 1 = 10

If the middle number ends in at least 5 zeros then the smallest number for the sum to be a cube is additional of 6 zeroes:

x = 9,000,000

x - 1 = 8,999,999

x + 1 = 9,000,001

sum = 27,000,000 = 300^3