Vol. 7, No. 8, 2013

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

Dinesh S. Thakur

Vol. 7 (2013), No. 8, 1841–1848

We consider an analog, when is replaced by Fq[t], of Wilson primes, namely the primes satisfying Wilson’s congruence (p 1)! 1 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 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 p1. 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.

Dedicated to Barry Mazur on his 75th birthday

Wilson prime, arithmetic derivative, Fermat quotient
Mathematical Subject Classification 2010
Primary: 11T55
Secondary: 11A41, 11N05, 11N69, 11A07
Received: 9 May 2012
Revised: 10 September 2012
Accepted: 31 October 2012
Published: 24 November 2013
Dinesh S. Thakur
1013 Hylan Building
Department of Mathematics
University of Rochester
RC Box 270138
Rochester, NY 14627
United States