MathNet 2022-part-04 — 197 problems (EN–VI) Question 138

Let the sequence $a_i$ be defined in the following way: $a_1 = m \in \mathbb{Z}_+$ and inductively $a_{i+1} = a_i + \lfloor \sqrt{a_i} \rfloor$. Prove that the sequence $a_i$ contains infinitely many perfect squares.