#### Vol. 14, No. 1, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Strongly nonzero points and elliptic pseudoprimes

### Liljana Babinkostova, Dylan Fillmore, Philip Lamkin, Alice Lin and Calvin L. Yost-Wolff

Vol. 14 (2021), No. 1, 65–88
##### Abstract

Efficiently distinguishing prime and composite numbers is one of the fundamental problems in number theory. A Fermat pseudoprime is a composite number $N$ which satisfies Fermat’s little theorem for a specific base $b$: ${b}^{N-1}\equiv 1\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}N$. A Carmichael number $N$ is a Fermat pseudoprime for all $b$ with $gcd\left(b,N\right)=1$. D. Gordon (1987) introduced analogues of Fermat pseudoprimes and Carmichael numbers for elliptic curves with complex multiplication (CM): elliptic pseudoprimes, strong elliptic pseudoprimes and elliptic Carmichael numbers. It has previously been shown that no CM curve has a strong elliptic Carmichael number. We give bounds on the fraction of points on a curve for which a fixed composite number $N$ can be a strong elliptic pseudoprime. J. Silverman (2012) extended Gordon’s notion of elliptic pseudoprimes and elliptic Carmichael numbers to arbitrary elliptic curves. We provide probabilistic bounds for whether a fixed composite number $N$ is an elliptic Carmichael number for a randomly chosen elliptic curve.

##### Keywords
elliptic curves, pseudoprimes, strongly nonzero elliptic pseudoprimes, elliptic Carmichael numbers
##### Mathematical Subject Classification 2010
Primary: 14H52, 14K22
Secondary: 11N25, 11G07, 11G20, 11B99
##### Supplementary material

Proofs of Theorems 4.7 and 4.20