RMM 2026 Problem 4

For any positive integer $m$, let $\varphi(m)$ be the number of positive integers less than or equal to $m$ and coprime to $m$. Define $\varphi_0(m) = m$ and, for each positive integer $k$, $\varphi_k(m) = \varphi(\varphi_{k-1}(m))$. For any integer $n \geq 3$, prove that

$$\varphi_0(2^n - 3) \cdot \varphi_1(2^n - 3) \cdot \varphi_2(2^n - 3) \cdot \ldots \cdot \varphi_n(2^n - 3)$$

has at most $n$ distinct prime divisors.