On Types of Elliptic Pseudoprimes

TL;DR

This paper extends the concept of elliptic pseudoprimes to include Euler-Jacobi and strong variants, exploring their properties, relationships, and density, especially focusing on elliptic Korselt numbers and anomalous primes.

Contribution

It introduces new types of elliptic Carmichael numbers and proves a conjecture about their density among elliptic Korselt numbers of Type I.

Findings

Defined Euler Elliptic Carmichael numbers and strong elliptic Carmichael numbers.

Established relationships among various elliptic pseudoprimes and Korselt numbers.

Proved a conjecture on the density of elliptic Korselt numbers of Type I that are products of anomalous primes.

Abstract

We generalize the notions of elliptic pseudoprimes and elliptic Carmichael numbers introduced by Silverman to analogues of Euler-Jacobi and strong pseudoprimes. We investigate the relationships among Euler Elliptic Carmichael numbers , strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I: The former two of these are introduced in this paper, and the latter two of these were introduced by Mazur (1973) and Silverman (2012) respectively. In particular, we expand upon a previous work of Babinkostova et al. by proving a conjecture about the density of certain elliptic Korselt numbers of Type I that are products of anomalous primes.