Article ID Journal Published Year Pages File Type
5772709 Journal of Number Theory 2017 15 Pages PDF
Abstract
In 1994, D. Thakur introduced the notion of Wieferich primes for the Carlitz module, hereafter called c-Wieferich primes. At almost the same time, L. Denis proved the Carlitz module analogue of the famous Fermat's Last Theorem. In this article, we relate Thakur's definition of c-Wieferich primes to Denis' result and state the necessary and sufficient condition for a monic irreducible (prime) polynomial P in Fq[T] to be c-Wieferich. We use this condition to give another proof for infinitude of c-Wieferich primes in F2[T] and in addition construct two algorithms for computing c-Wieferich primes. With the help of the SAGE software, we compute several examples of c-Wieferich primes for the rings Fq[T], where q∈{3,5,7,11,13,19,29,37}. Lastly, we unconditionally prove infinitude of non-c-Wieferich primes in Fq[T] for q>2.
Keywords
Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
Authors
,