Remote Access Mathematics of Computation
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $n|{\psi _{n + 1}}(P)$ 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 $n|{\psi _{n + 1}}({P_i})$, $1 \leq i \leq r$. For $r \geq 2$, we prove ${N_\Gamma }(x) \ll x\exp ( - c\sqrt {(\log x)(\log \log x))}$ for some positive constant c.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11G05, 11A51, 11Y11

Retrieve articles in all journals with MSC: 11G05, 11A51, 11Y11

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society