A high school Olympiad problem

Let $p(n)$ denote the largest prime factor of $n$. Prove that there are infinitely many $n$ such that $$p(n)<p(n+1)<p(n+2).$$

Edit:

My solution: Choose $n=\prod_{i=1}^{k}p_i$, product of the first $k$ primes. This means that $n+1=\prod_{i=1}^{k}p_i+1$ is either a larger prime or has a prime factor larger than that of all factors of $n$. Still not sure about $n+2$.

Solution 1:

The problem is solved in the paper by Erdos and Pomerance "On the largest prime factors of $n$ and $n+1$".

The proof in the paper is quite simple:

Fix a prime $l>2$ and define $k_0=\inf\{k\mid p(l^{2^k}+1)>l\}<\infty$. Then we have $p(n)<p(n+1)<p(n+2)$ when $n=l^{2^{k_0}}-1$.