For a positive integer $n$, let $p(n)$ the largest prime divisor of $n$. Show that there exist infinitely many positive integers m such that $p(m − 1) < p(m) < p(m + 1)$.
Let $q$ be odd prime and $a<b$.
$q^{2^b} - 1 =(q-1)(q+1)(q^2+1)...(q^{2^{b-1}}+1)$
$q^{2^{a}}+1 $ is a divisor of $ q^{2^{b}}-1 $
I don't know how to get the required inequality.
