Power savings for counting solutions to polynomial-factorial equations

TL;DR

This paper establishes a power-saving upper bound on the number of integer solutions to factorial-polynomial equations, advancing understanding of a classical problem by employing Diophantine and Padé approximation techniques.

Contribution

It provides the first power-saving bound for solutions to factorial-polynomial equations, improving upon the previous o(N) bound and addressing a longstanding mathematical question.

Findings

Proves a power-saving upper bound on solutions to n! = P(x).

Applies techniques of Diophantine and Padé approximation.

Addresses a century-old problem of Brocard and Ramanujan.

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 $n\leq N$ 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 Pad\'e approximation.