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INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

Part of: Multiplicative number theory Computational number theory Elementary number theory

Published online by Cambridge University Press:  25 April 2012

AARON EKSTROM ,
CARL POMERANCE  and
DINESH S. THAKUR
AARON EKSTROM
Affiliation:
8505 E. Ocotillo Dr., Tucson AZ 85750, USA (email: DoctorX198@gmail.com)
CARL POMERANCE*
Affiliation:
Mathematics Department, Dartmouth College, Hanover, NH 03755, USA (email: carl.pomerance@dartmouth.edu)
DINESH S. THAKUR
Affiliation:
Mathematics Department, University of Arizona, Tucson, AZ 85721, USA (email: thakur@math.arizona.edu)
*
✉For correspondence; e-mail: carl.pomerance@dartmouth.edu
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Abstract

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In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

Keywords

probable primespseudo-primesprimality testselliptic curvescomplex multiplication

MSC classification

Secondary: 11A51: Factorization; primality 11Y11: Primality 11N13: Primes in progressions 11N36: Applications of sieve methods 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 1 , February 2012 , pp. 45 - 60
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The second author was supported in part by NSF grant DMS-1001180. The third author was supported in part by NSA grant H98230-10-1-0200.

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