Differential characterization of Wilson primes for 𝔽q[t]

Thakur, Dinesh

doi:10.2140/ant.2013.7.1841

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

Dinesh S. Thakur

Vol. 7 (2013), No. 8, 1841–1848

DOI: 10.2140/ant.2013.7.1841

Abstract

We consider an analog, when $ℤ$ is replaced by $F_{q} [t]$ , of Wilson primes, namely the primes satisfying Wilson’s congruence $(p - 1)! \equiv - 1$ 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 $℘$ of $F_{q} [t]$ 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 $℘^{p - 1}$ . 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.

Dedicated to Barry Mazur on his 75th birthday

Keywords

Wilson prime, arithmetic derivative, Fermat quotient

Mathematical Subject Classification 2010

Primary: 11T55

Secondary: 11A41, 11N05, 11N69, 11A07

Milestones

Received: 9 May 2012

Revised: 10 September 2012

Accepted: 31 October 2012

Published: 24 November 2013

Authors

Dinesh S. Thakur
1013 Hylan Building Department of Mathematics University of Rochester RC Box 270138 Rochester, NY 14627 United States

Vol. 7, No. 8, 2013