%0 Journal Article %T Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers %A Vladimir Shevelev %A Gilberto Garc¨ªa-Pulgar¨ªn %A Juan Miguel Vel¨¢squez-Soto %A John H. Castillo %J Mathematics %D 2012 %I arXiv %X 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 $b$. %U http://arxiv.org/abs/1206.0606v1