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 denote the *n*th division polynomial, and let *P* be a rational point of *E* of infinite order. A natural number *n* is called an *elliptic pseudoprime* if and *n* is composite. Let denote the number of elliptic pseudoprimes up to *x*. We show that . More generally, if are *r* independent rational points of *E* which have infinite order, and is the subgroup generated by them, denote by the number of composite satisfying , . For , we prove for some positive constant *c*.

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0970701-9

Article copyright:
© Copyright 1989
American Mathematical Society