For an elliptic curve E and fixed integer r , Lang and Trotter have conjectured an asymptotic estimate for the number of primes p {600} x such that the trace of Frobenius a p ( E ) = r . Using similar heuristic reasoning, Koblitz has conjectured an asymptotic estimate for the number of primes p {600} x such that the order of the group of points of E over the finite field [Special characters omitted.] is also prime. These estimates have been proven correct for elliptic curves "on average"; however, beyond this the conjectures both remain open.  In this thesis, we combine the condition of Lang and Trotter with that of Koblitz to conjecture an asymptotic for the number of primes p {600} x such that both