Daniel Berend

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 ≤ r ≤ 3 (although it undoubtedly should be). We also study the equation P(x) = H_n, where (H_n) is one of several other "highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.