Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields

TL;DR

This paper develops analogues of Mersenne primes, Wieferich primes, and related primes within the framework of Carlitz modules, extending classical number theory concepts to function fields and establishing new results about their properties.

Contribution

It introduces Carlitz module analogues of Mersenne primes and related primes, proving classical results in this new setting and showing the infinitude of composite Mersenne numbers.

Findings

Carlitz module analogues of Mersenne primes are constructed.

Classical results about Mersenne primes are extended to the Carlitz module context.

Infinitely many composite Mersenne numbers are demonstrated.

Abstract

In this paper, we introduce a Carlitz module analogue of Mersenne primes, and prove Carlitz module analogues of several classical results concerning Mersenne primes. In contrast to the classical case, we can show that there are infinitely many composite Mersenne numbers. We also study the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module context that were first introduced by Dinesh Thakur.