The Lang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx$

as x −→ ∞, where CE,r is a specific non-negative constant. The Hardy-Littlewood Conjecture gives a similar asymptotic formula as above for the number of primes of the form ax + bx + c. We establish a relationship between the Hardy-Littlewood Conjecture and the Lang-Trotter Conjecture for the elliptic curve y = x+Dx. We show that the Hardy-Littlewood Conjecture implies the Lang-Trotter Conjecture for y = x + Dx. Conversely, if the Lang-Trotter Conjecture holds for some D and 2r (for y = x + Dx, p ∤ D, ap is always even) with positive constant CE,2r, then the polynomial x + r represents infinitely many primes. For a prime p, if ap = 2r, then p is necessarily of the form x 2 + r. Fixing r and D, and assuming that the Hardy-Littlewood Conjecture holds, we obtain the density of the primes with ap = 2r inside the set of primes of the form x 2 + r. In some cases, the density is 1/4, which is a natural expectation, but it fails to be true for all D. In particular, we give a full list of D and r when there is no prime p for ap = 2r.

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