## Abstract

We introduce a new class of pseudoprimes, that we call "overpseudoprimes to base b", which is a subclass of the strong pseudoprimes to base b. Letting |b|_{n} denote the multiplicative order of b modulo n, we show that a composite number n is an overpseu- doprime 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 overpseudoprimes to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well-known numbers are "primover to base b"; i.e., they are primes or overpseudoprimes to base b.

Original language | English |
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Journal | Journal of Integer Sequences |

Volume | 15 |

Issue number | 7 |

State | Published - 8 Sep 2012 |

## Keywords

- Cyclotomic cosets of
- Mersenne numbers
- Modulo n
- Order of
- Overpseudoprime, Wieferich prime
- Poulet pseudoprime
- Super-poulet pseudoprime

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics