Differential characterization of Wilson primes for $\mathbb{F}_q[t]$

Abstract

We consider an analog, when $\mathbb{Z}$ is replaced by ${\mathbb{F}}_q\left[t\right]$, of Wilson primes, namely the primes satisfying Wilson’s congruence $\left(p-1\right)!\equiv -1$ to modulus $p^2$ rather than the usual prime modulus $p$. We fully characterize these primes by connecting these or higher power congruences to other fundamental quantities such as higher derivatives and higher difference quotients as well as higher Fermat quotients. For example, in characteristic $p>2$, we show that a prime $\wp$ of ${\mathbb{F}}_q\left[t\right]$ is a Wilson prime if and only if its second derivative with respect to $t$ is $0$ and in this case, further, that the congruence holds automatically modulo ${\wp }^{p-1}$. For $p=2$, the power $p-1$ is replaced by $4-1=3$. For every $q$, we show that there are infinitely many such primes.