Elliptic pseudoprime

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 ${\displaystyle \mathbb {Q} {\big (}{\sqrt {-d}}{\big )}}${\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

${\displaystyle X/\exp((1/3)\log X\log \log \log X/\log \log X)\ .}${\displaystyle X/\exp((1/3)\log X\log \log \log X/\log \log X)\ .}

• 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.

• Weisstein, Eric W. "Elliptic Pseudoprime". MathWorld .

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