Problem

Source: USAJMO 2022/1

Tags: USA, USAJMO, number theory, arithmetic sequence, geometric sequence, Sequence, USA(J)MO



For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties? $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$; $a_2 - a_1$ is not divisible by $m$. Holden Mui