Overpseudoprimes, Mersenne Numbers and Wieferich primes

TL;DR

This paper introduces overpseudoprimes, a new class of pseudoprimes characterized by invariant multiplicative order, and demonstrates their properties and relationships with Mersenne numbers and Wieferich primes.

Contribution

It defines overpseudoprimes, establishes their characterization via multiplicative order invariance, and links them to Mersenne numbers and Wieferich primes, extending the understanding of pseudoprime classes.

Findings

All composite Mersenne numbers are overpseudoprimes.

Squares of Wieferich primes are overpseudoprimes.

Every overpseudoprime is a strong pseudoprime of the same base.

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.