Primitive Divisors of Certain Elliptic Divisibility Sequences

Let E : y 2 = x 3 + D be an elliptic curve, where D is an integer that contains no primes p with 6 | ord p D. For a nontorsion rational point P on E, write x(nP) = A n (P)/B 2 n (P) in lowest terms. We prove that for the sequence , the term has a primitive divisor for all m ≥ 3. As an application, we give a new method for solving the Diophantine equation y 2 = x 3 + d n under certain conditions.