A note on Carlitz Wieferich primes

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• 作者：Alex Samuel Bamunoba^a ; ^b ; ^{bamunoba@aims.ac.za}
• 关键词：11A99 ; 11G09
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：343-357
• 全文大小：378 K
• 卷排序：174

文摘

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]Fq[T] to be c-Wieferich. We use this condition to give another proof for infinitude of c-Wieferich primes in F2[T]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]Fq[T], where q∈{3,5,7,11,13,19,29,37}q∈{3,5,7,11,13,19,29,37}. Lastly, we unconditionally prove infinitude of non-c-Wieferich primes in Fq[T]Fq[T] for q>2q>2.