The connection between the multiplicative and additive structures of an arbitrary integer is one of the most intriguing problems in number theory. It is in this context that we explore the problem of identifying those consecutive integers which are divisible by a power of their largest prime factor. For instance, letting $P(n)$ stand for the largest prime factor of $n$, then the number $n=1294298$ is the smallest integer which is such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2$. No one has yet found an integer $n$ such that $P(n+i)^2$ divides $n+i$ for $i=0,1,2,3$. Why is that? In this talk, we will provide an answer to this question and explore similar problems.

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Last updated: 24 Jan 2020