Vol. 14, No. 1, 2021

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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: bN1 1 mod N. A Carmichael number N is a Fermat pseudoprime for all b with gcd(b,N) = 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

Milestones
Received: 1 October 2019
Revised: 12 August 2020
Accepted: 29 September 2020
Published: 4 March 2021

Communicated by Kenneth S. Berenhaut
Authors
Liljana Babinkostova
Department of Mathematics
Boise State University
Boise, ID
United States
Dylan Fillmore
Department of Mathematics
University of South Carolina
Columbia, SC
United States
Philip Lamkin
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA
United States
Alice Lin
Department of Mathematics
Princeton University
Princeton, NJ
United States
Calvin L. Yost-Wolff
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States