Question 169
Prove that there exists a positive number $C$ such that the following statement holds: for any infinite arithmetic progression $a_1, a_2, a_3, \dots$ of positive integers, if the greatest common divisor of $a_1$ and $a_2$ is square-free, then there exists some positive integer $m \le C \cdot a_2^2$ such that $a_m$ is square-free.
*Remark: We call a positive integer $N$ square-free if it is not divisible by any square that is strictly larger than 1.*
*Remark: We call a positive integer $N$ square-free if it is not divisible by any square that is strictly larger than 1.*