Consider a polynomial $P(X)\in\mathbb Z[X]$. Is it true that $P(N)$ divides $N!$ for infinitely many integer $N$?


This question is motivated by the special case where $P(X) = X^2 + 1$ that appeared in a math olympiad.


I was wondering if anyone can point me to references of this question.



Integral polynomials dividing N!
In general this is an open problem. A closely related problem (essentially equivalent) is to ask for the values $P(n)$ for $n$ of size $X$ to be $X$ smooth (i.e. composed only of primes below $X$). This is known for quadratic polynomials, but already open for general cubic irreducible polynomials. For some special polynomials (e.g. those of the form $ax^d+b$) such results are known, thanks to Schinzel, Balog and Wooley etc. For a description of related results, and more references, see this paper of [Dartyge, Martin and Tenenbaum](http://www.akademiai.com/doi/pdf/10.1023/A%3A1015237700066). 


