  Let $E$ be an elliptic curve over $\mathbb{Q}.$ Let $a_p$ denote the trace of
the Frobenius endomorphism at a rational prime $p$. For a fixed integer $r,$
define the prime-counting function as $\pi_{E,r}(x):=\sum_{p\leq x,p\nmid
\Delta_E,a_p=r}1$. The Lang-Trotter Conjecture predicts that
$$\pi_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o(\frac{\sqrt{x}}{{\rm
log}x})$$ as $x\longrightarrow \infty,$ where $C_{E,r}$ 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 $ax^2+bx+c$.
We establish a relationship between the Hardy-Littlewood Conjecture and the
Lang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx.$ We show that the
Hardy-Littlewood Conjecture implies the Lang-Trotter Conjecture for
$y^2=x^3+Dx.$ Conversely, if the Lang-Trotter Conjecture holds for some $D$ and
$2r$ (for $y^2=x^3+Dx, p\nmid D, a_p$ is always even) with positive constant
$C_{E,2r},$ then the polynomial $x^2+r^2$ represents infinitely many primes.
For a prime $p$, if $a_p=2r$, then $p$ is necessarily of the form $x^2+r^2$.
Fixing $r$ and $D$, and assuming that the Hardy-Littlewood Conjecture holds, we
obtain the density of the primes with $a_p=2r$ inside the set of primes of the
form $x^2+r^2$. 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 $a_p=2r$.
