Anomalous primes of the elliptic curve E D : y 2 *x 3 +D

Let D ∈ Z be an integer that is neither a square nor a cube in Q ( − 3 ) , and let E D be the elliptic curve defined byy 2 = x 3 + D . Mazur conjectured that the number of anomalous primes less than N should be given asymptotically by c N / log N ( c is a positive constant), and in particular there should be infinitely many anomalous primes for E D . We show that the Hardy–Littlewood conjecture implies the Mazur conjecture, except for D = 80 d 6 , where 0 ≠ d ∈ Z [ ( 1 + − 3 ) / 2 ] withd 6 ∈ Z . Conversely, if the Mazur conjecture holds for some D , then the polynomial 12 x 2 + 18 x + 7 represents infinitely many primes. All anomalous primes belong to the quadratic progression q ( h ) = 1 4 ( 1 + 3 h 2 ) . Assuming the Hardy–Littlewood conjecture, we obtain the density of the anomalous primes in the primes in q ( h ) for any D . The density is1 6in some cases, as Mazur had conjectured, but it fails to be true for all D . Our results are more general. In fact, we will consider all primes of six types which belong to q ( h ) , not just anomalous primes. The density results are established for all these primes. We also discuss the Lang–Trotter conjecture for E D .