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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
##### Keywords:
Carmichael number; pseudoprime; elliptic curve
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