$q$-pseudoprimality: A natural generalization of strong pseudoprimality

TL;DR

This paper introduces the concept of q-pseudoprimality as a generalization of strong pseudoprimes, offering new insights into pseudoprime classifications and their properties, including the absence of Carmichael-like numbers in this context.

Contribution

It defines q-pseudoprimes to base b, relates them to Midy's numbers, and analyzes their properties, including the non-existence of Carmichael analogs.

Findings

Defined q-pseudoprimes to base b

Connected q-pseudoprimes with Midy's numbers

Proved no Carmichael-like numbers exist for q-pseudoprimes

Abstract

In this work we present a natural generalization of strong pseudoprime to base $b$, which we have called $q$-pseudoprime to base $b$. It allows us to present another way to define a Midy's number to base $b$ (overpseudoprime to base $b$). Besides, we count the bases $b$ such that $N$ is a $q$-probable prime base $b$ and those ones such that $N$ is a Midy's number to base $b$. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to $q$-probable prime to base $b$ as with the concept of strong pseudoprimes to base $b$.