MathNet 2005 — Part 03 Question 150

Let $a_{1} = 3$, and for $n \geq 1$, let $a_{n+1} = (n+1) a_{n} - n$. Find the smallest $m \geq 2005$ such that $a_{m+1} - 1 \mid a_{m}^{2} - 1$.