Brocard-Ramanujan problem for polynomials over finite fields

Abstract

The Brocard-Ramanujan problem is an unsolved number theory problem to find integer solutions $(x,n)$ to $x^2-1=n!$. In this paper, we consider this problem over polynomial rings $F_q\left[T\right]$, where $F_q$ is a finite field with q elements. We find all solutions to the equation $X^2-1={\Pi }_C\left(n\right)$, where ${\Pi }_C\left(n\right)$ denotes the Carlitz factorial. More precisely, we characterize all solutions and prove that there are infinitely many solutions if and only if $F_q$ is an extension of $F_4$. This characterization is achieved without using the Mason-Stothers theorem, analogous to the abc conjecture for integers.