On the elliptic curve y 2 = x 3-2rDx and factoring integers

Let D = pq be the product of two distinct odd primes. Assuming the parity conjecture, we construct infinitely many r a (c) 3/4 1 such that E (2rD) : y (2) = x (3) -2rDx has conjectural rank one and v (p) (x([k]Q)) not equal v (q) (x([k]Q)) for any odd integer k, where Q is the generator of the free part of E(a"e). Furthermore, under the generalized Riemann hypothesis, the minimal value of r is less than c log(4) D for some absolute constant c. As a corollary, one can factor D by computing the generator Q.