Question 139
Let $m$ be a positive integer, let $p$ be a prime, let $a_1 = 8p^m$, and let $a_n = (n+1)^{a_{n-1}/n}$, $n = 2, 3, \dots$. Determine the primes $p$ for which the products $a_n\left(1 - \frac{1}{a_1}\right)\left(1 - \frac{1}{a_2}\right)\dots\left(1 - \frac{1}{a_n}\right)$, $n = 1, 2, 3, \dots$, are all integral.