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2013 Differential characterization of Wilson primes for $\mathbb{F}_q[t]$
Dinesh Thakur
Algebra Number Theory 7(8): 1841-1848 (2013). DOI: 10.2140/ant.2013.7.1841


We consider an analog, when is replaced by Fq[t], of Wilson primes, namely the primes satisfying Wilson’s congruence (p1)!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 p1 is replaced by 41=3. For every q, we show that there are infinitely many such primes.


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Dinesh Thakur. "Differential characterization of Wilson primes for $\mathbb{F}_q[t]$." Algebra Number Theory 7 (8) 1841 - 1848, 2013.


Received: 9 May 2012; Revised: 10 September 2012; Accepted: 31 October 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1291.11013
MathSciNet: MR3134036
Digital Object Identifier: 10.2140/ant.2013.7.1841

Primary: 11T55
Secondary: 11A07 , 11A41 , 11N05 , 11N69

Keywords: arithmetic derivative , Fermat quotient , Wilson prime

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.7 • No. 8 • 2013
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