  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.
