In number theory, a pseudoprime is called an elliptic pseudoprime for (E, P), where E is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in Q ( − d ) {\displaystyle \mathbb {Q} {\big (}{\sqrt {-d}}{\big )}} , having equation y2 = x3 + ax + b with a, b integers, P being a point on E and n a natural number such that the Jacobi symbol (−d | n) = −1, if (n + 1)P ≡ 0 (mod n). The number of elliptic pseudoprimes less than X is bounded above, for large X, by
X / exp ( ( 1 / 3 ) log X log log log X / log log X ) . {\displaystyle X/\exp((1/3)\log X\log \log \log X/\log \log X)\ .}
References
Gordon, Daniel M.; Pomerance, Carl (1991). "The distribution of Lucas and elliptic pseudoprimes". Mathematics of Computation. 57 (196): 825–838. doi:10.2307/2938720. JSTOR 2938720. Zbl 0774.11074.
External links Weisstein, Eric W. "Elliptic Pseudoprime". MathWorld.