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Let $E$ be an elliptic curve over $\mathbb{Q}.$ Let $a_p$ denote the trace ofthe 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$$\pi_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o(\frac{\sqrt{x}}{{\rmlog}x})$$ as $x\longrightarrow \infty,$ where $C_{E,r}$ is a specificnon-negative constant. The Hardy-Littlewood Conjecture gives a similarasymptotic 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 theLang-Trotter Conjecture for the elliptic curve $y^2=x^3+Dx.$ We show that theHardy-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, weobtain the density of the primes with $a_p=2r$ inside the set of primes of theform $x^2+r^2$. In some cases, the density is $1/4$, which is a naturalexpectation, but it fails to be true for all $D$. In particular, we give a fulllist of $D$ and $r$ when there is no prime $p$ for $a_p=2r$.

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