Let us say that three consecutive positive integers $(m-1,m,m+1)$ have a big prime factor if the largest prime factor $p$ of $N:=(m-1)m(m+1)$ satisfies $e^p>N$.
I ckecked that it is true for all $m<10^8$, except for $m=3,9,15,49,55,99,351,441,2431$ where $p=3,5,7,7,11,11,13,17,19$, respectively.
Question: Is it true for all $m>2431$?
Application: A positive answer would solve a problem in Section D25 of Richard Guy's Unsolved Problems in Number Theory.
Simmons notes that $n! = (m - 1)m(m + 1)$ for $(m, n) = (2, 3)$, $(3, 4)$, $(5, 5)$, and $(9, 6)$, and asks if there are other solutions. More generally he asks if there are any other solutions of $n! + x = x^k$. This is a variation on the question of asking for $n!$ to be the product of $k$ consecutive integers in a nontrivial way ($k \ne n + 1 - j!$). Compare B23.
Note that $n! > e^n$ iff $n \ge 6$. But I checked for $m \le 2431$ that $n! = (m-1)m(m+1)$ iff $$(m,n) = (2,3), (3,4), (5,5), (9,6).$$ So for $m>2431$, if $e^p>N$ and $N = n!$, then $p! \ge n!$, but $p<n$ because $p$ is a prime factor of $n!$ (and $n>2$), contradiction.
Our question for three consecutive positive integers can be generalized to $k \ge 3$ consecutive positive integers, where the bound $e^p$ could be optimized. Now for $k=2$ it is not clear that $e^p$ (or even $p!$) works because I found large exceptions.
sage: m=123200 sage: factor(m*(m+1)) 2^6 * 3^6 * 5^2 * 7 * 11 * 13^2 sage: factorial(13)>m*(m+1) False sage: m=2697695 sage: factor(m*(m+1)) 2^5 * 3^2 * 5 * 7^3 * 11^2 * 13 * 17 * 19 * 29 sage: ceil(exp(29))>m*(m+1) False
On the contrary, for $k=2$ we can wonder whether, for a given prime $p>2$, there are infinitely many $m$ such that the biggest prime factor of $m(m+1)$ is $\le p$. The case $p=3$ reduces to the existence of infinitely many couple of non-negative integers $(a,b)$ with $\lvert 2^a-3^b\rvert = 1$, but it is false (see why at distance between powers of 2 and powers of 3). Now what if $p$ is large enough? In other words:
Bonus question: Is there an integer $n$ such that the set of integers $m(m+1)$ whose prime factors are less than $n$, is infinite?