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  • INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

  • Part of
  • AARON EKSTROM, CARL POMERANCE, DINESH S. THAKUR
  • Journal:
    Journal of the Australian Mathematical Society / Volume 92 / Issue 1 / February 2012
    Published online by Cambridge University Press:
    25 April 2012, pp. 45-60
    Print publication:
    February 2012
    • Article
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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.