Elliptic Carmichael Numbers and Elliptic Korselt Criteria

TL;DR

This paper introduces elliptic Carmichael numbers and provides criteria similar to Korselt's for identifying them, focusing on numbers with two prime factors and their properties related to elliptic curves.

Contribution

It develops two elliptic Korselt criteria for elliptic Carmichael numbers and analyzes their structure, especially for numbers with two prime factors.

Findings

Elliptic Carmichael numbers can be characterized using Korselt-like criteria.

The paper provides conditions for elliptic Carmichael numbers of the form pq.

Analysis of elliptic Carmichael numbers with two prime factors (pq).

Abstract

Let E/Q be an elliptic curve, let L(E,s)=\sum a_n/n^s be the L-series of E/Q, and let P be a point in E(Q). An integer n > 2 having at least two distinct prime factors will be be called an elliptic pseudoprime for (E,P) if E has good reduction at all primes dividing n and (n+1-a_n)P = 0 (mod n). Then n is an elliptic Carmichael number for E if n is an elliptic pseudoprime for every P in E(Z/nZ). In this note we describe two elliptic analogues of Korselt's criterion for Carmichael numbers, and we analyze elliptic Carmichael numbers of the form pq.