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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
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Abstract |
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Efficiently distinguishing prime and composite numbers is one of the fundamental problems in number theory. A Fermat pseudoprime is a composite number which satisfies Fermat’s little theorem for a specific base : . A Carmichael number is a Fermat pseudoprime for all with . 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 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 is an elliptic Carmichael number for a randomly chosen elliptic curve. |
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Keywords
elliptic curves, pseudoprimes, strongly nonzero elliptic
pseudoprimes, elliptic Carmichael numbers
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Mathematical Subject Classification 2010
Primary: 14H52, 14K22
Secondary: 11N25, 11G07, 11G20, 11B99
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Supplementary material |
Milestones
Received: 1 October 2019
Revised: 12 August 2020
Accepted: 29 September 2020
Published: 4 March 2021
Communicated by Kenneth S. Berenhaut
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Authors |
| Liljana Babinkostova | |
| Department of Mathematics Boise State University Boise, ID United States |
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| Dylan Fillmore | |
| Department of Mathematics University of South Carolina Columbia, SC United States |
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| Philip Lamkin | |
| Department of Mathematics Carnegie Mellon University Pittsburgh, PA United States |
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| Alice Lin | |
| Department of Mathematics Princeton University Princeton, NJ United States |
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| Calvin L. Yost-Wolff | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
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