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Elliptic Carmichael numbers and elliptic Korselt criteria. (English) Zbl 1304.11047
Summary: Let \(E/\mathbb Q\) be an elliptic curve, let \(L(E,s)=\sum a_n/n^s\) be the \(L\)-series of \(E/\mathbb Q\), and let \(P\) be a point in \(E(\mathbb Q)\). An integer \(n > 2\) having at least two distinct prime factors will be be called an elliptic pseudoprime for \((E,P)\) if \(E\) has good reduction at all primes dividing \(n\) and \((n+1-a_n)P = 0\pmod n\). Then \(n\) is an elliptic Carmichael number for \(E\) if \(n\) is an elliptic pseudoprime for every \(P\) in \(E(\mathbb Z/n\mathbb Z)\). In this note we describe two elliptic analogues of Korselt’s criterion for Carmichael numbers, and we analyze elliptic Carmichael numbers of the form \(pq\).

MSC:
11G05 Elliptic curves over global fields
11Y11 Primality
Software:
PARI/GP
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