Overpseudoprimes, Mersenne Numbers and Wieferich primes

Abstract

We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value of h(n) is invariant of all divisors d>1 of n. In particular, we prove that all composite Mersenne numbers 2^p-1,where p is prime, and squares of Wieferich primes are overpseudoprimes. We give also a generalization of the results on arbitrary base a>1 and prove that every overpseudoprime is strong pseudoprime of the same base.

Adding of the second proof of Theorem 2

Cited by in corpus (2)

• Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers
• Pseudoprimes stronger than strong pseudoprimes