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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Elliptic pseudoprimes

Authors: I. Miyamoto and M. Ram Murty
Journal: Math. Comp. 53 (1989), 415-430
MSC: Primary 11G05; Secondary 11A51, 11Y11
MathSciNet review: 970701
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Abstract: Let E be an elliptic curve over Q with complex multiplication by an order in an imaginary quadratic field. Let $ {\psi _n}$ denote the nth division polynomial, and let P be a rational point of E of infinite order. A natural number n is called an elliptic pseudoprime if $ n\vert{\psi _{n + 1}}(P)$ and n is composite. Let $ N(x)$ denote the number of elliptic pseudoprimes up to x. We show that $ N(x) \ll x{(\log \log x)^{7/2}}/{(\log x)^{3/2}}$. More generally, if $ {P_1}, \ldots ,{P_r}$ are r independent rational points of E which have infinite order, and $ \Gamma $ is the subgroup generated by them, denote by $ {N_\Gamma }(x)$ the number of composite $ n \leq x$ satisfying $ n\vert{\psi _{n + 1}}({P_i})$, $ 1 \leq i \leq r$. For $ r \geq 2$, we prove $ {N_\Gamma }(x) \ll x\exp ( - c\sqrt {(\log x)(\log \log x))} $ for some positive constant c.

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Article copyright: © Copyright 1989 American Mathematical Society