Power savings for counting solutions to polynomial-factorial equations

Abstract

Let P be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions n ≤ N to n! = P (x) which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is o(N ). The proof uses techniques of Diophantine and Padé approximation.2010 Mathematics Subject Classification. 11D45.