KöMaL 2013 A578

For every integer $n \ge$2 let $P$($n$) be the product of all expressions of the form $\pm\sqrt1 \pm\sqrt2 \pm\sqrt3 \pm\ldots \pm\sqrt{n}$ where the signs of the terms are chosen arbitrarily. ($a$) Prove that $P$($n$) is a positive integer. ($b$) Prove that for all $\varepsilon$>0 there exists an $n_0$ such that for every $n$>$n_0$ the largest prime divisor of $P$($n$) is smaller than $2^{2^{\varepsilon n}}$.