MathNet 2019-part-07 — 184 problems (EN–VI) Question 144

A sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by $a_{n+1} = a_n^2 + 2018$, where $a_1$ is some positive integer. Prove that in this sequence no more than one number can be a cube of a positive integer.