Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers

Abstract

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 $b$.

9 pages

References in corpus (2)

• Pseudoprimes stronger than strong pseudoprimes
• Process of "Primoverization" of Numbers of the Form a^n-1