MathNet 2023-part-03 — 191 problems (EN–VI) Question 175

Denote by $k!!$ the product $k \times (k-2) \times \cdots \times 1$ for any odd integer $k \ge 1$. Show that $(2^m - 1)!! - 1$ is divisible by $2^m$ for any integer $m \ge 3$.