Let $p(n)$ is the biggest prime divisor of $n$. Prove that, exist infinite $n \in N$ satify : $p(n) <p(n+1)<p(n+2)$
From page 320 of "On the largest prime factors of $n$ and $n+1$" by Paul Erdos and Carl Pomerance (Aequationes Mathematicae 17, 1978, pp. 311-321):
Suppose now $p$ is an odd prime and
$$k_0=\inf\{k:P(p^{2^k}+1)\gt p\}$$
(note that $P(p^{2^{k_0}}+1)\equiv 1$ mod$(2^{k_0+1})$, so $k_0\lt\infty$). Then
$$P(p^{2^{k_0}}-1)\lt P(p^{2^{k_0}})\lt P(p^{2^{k_0}}+1)$$
Remark: They go on to say:
On the other hand, we cannot find infinitely many $n$ for which
$$P(n)\gt P(n+1)\gt P(n+2)$$
but perhaps we overlook a simple proof.