A note on Carlitz Wieferich primes

Abstract

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 Pin 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-Wieferichprimes in Fq[T]for q>2.