The Non-existence of Universal Carmichael Numbers

Jan-Christoph Schlage-Puchta ()
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Jan-Christoph Schlage-Puchta: Universität Rostock, Institut für Mathematik

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 435-453 from Springer

Abstract: Abstract We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer n, which is not a prime power, is an elliptic Carmichael number for a random curve E with good reduction modulo n, is bounded above by $$\mathcal{O}(\log ^{-1}n)$$ . If we choose both n and E at random, the probability that n is E-Carmichael is bounded above by $$\mathcal{O}(n^{-1/8+\epsilon })$$ .

Keywords: Carmichael numbers; Elliptic curves; Pseudoprimes; Primary 11G20; Secondary 11N25, 11Y11 (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-28203-9_26

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DOI: 10.1007/978-3-319-28203-9_26

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