Problem

Source: 2021 Taiwan TST Round 3 Independent Study 1-N

Tags: number theory, ming



Let $a_1$, $a_2$, $a_3$, $\ldots$ be a sequence of positive integers such that $a_1=2021$ and $$\sqrt{a_{n+1}-a_n}=\lfloor \sqrt{a_n} \rfloor. $$Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence. Proposed by Li4, Tsung-Chen Chen, and Ming Hsiao.