Abstract
We consider an analog, when ⤠is replaced by Fq[t], of Wilson primes, namely the primes satisfying Wilsonâs congruence (pâ1)!âĄâ1 to modulus p2 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 â of Fq[t] 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 âpâ1. For p=2, the power pâ1 is replaced by 4â1=3. For every q, we show that there are infinitely many such primes.
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