TITLE: Is it true that for any two integers with the same least prime factor, there must be an integer with a higher lpf between them?
QUESTION [5 upvotes]: I'm sure something like this has been asked, but I can't seem to find it.
Let $a,b$ be two natural numbers such that $a<b$ and $\text{lpf}(a)=\text{lpf}(b)$. Is there always an $n\in\mathbb N$ such that $a<n<b$ and $\text{lpf}(n)>\text{lpf}(a)$?

REPLY [8 votes]: I am kind of astonished, but I've eventually discovered this to be false. There is a counterexample in the interval 724968762211953720363081773921156853174119094876349 to 724968762211953720363081773921156853174119094876611, which both have lpf of 131, with intervening values all lower.