Overpseudoprimes, Mersenne Numbers and Wieferich primes

Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers

We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $n$, we show that a composite $n$ is overpseudoprime if and only if $|b|_d$ is invariant for all divisors $d>1$ of $n$. In particular, we prove that all composite Mersenne numbers $2^{p}-1$, where $p$ is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base $b$.

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

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.

魏弗里希与梅森素数在函数域中的研究

Wieferich and Mersenne primes for function fields

本文研究了函数域中魏弗里希素数与梅森数的性质及其与费马方程的关系。