IMO 2026 Problem 6

Let $a_1, a_2, a_3, \ldots$ be an infinite sequence of positive integers greater than $1$. Suppose that for all positive integers $n$, the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that $\gcd(a_{n+1}, a_i) > 1$ for every $i = 1, 2, \ldots, n$. Prove that there exist positive integers $T$ and $L$ such that

$$a_{n+T} = a_n + L$$

for every positive integer $n$.

(Note that $\gcd(x, y)$ denotes the greatest common divisor of positive integers $x$ and $y$.)