Elliptic pseudoprime
# 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 $\mathbb {Q} {\big{\sqrt {-d}}{\big )}$, having equation $y^2$ = $x^3$ + 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

    $X/\exp((1/3)\log X\log \log \log X/\log \log X)\ .$
