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 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 &equiv; 0 (mod n).

The number of elliptic pseudoprimes less than X is bounded above, for large X, by


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