### abstract

- We study equations of the form P(x) = n! and show that for some classes of polynomials P the equation has only finitely many solutions. This is the case, say, if P is irreducible (of degree greater than 1) or has an irreducible factor of "relatively large" degree. This is also the case if the factorization of P contains some "large" power(s) of irreducible(s). For example, we can show that the equation $x^{r}(x + 1) = n!$ has only finitely many solutions for r ≥ 4, but not that this is the case for $1 \leq r \leq 3$ (although it undoubtedly should be). We also study the equation $P(x) = H_n$ , where (Hn) is one of several other "highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.