Anomalous primes of the elliptic curve $E_D : y^2 = x^3 + D$

It is open whether there exists a polynomial in one variable of degree $>1$  that represents infinitely many primes.  For example, at present, we do not know whether the polynomial $x^2+1$ represents infinitely many primes. The Hardy-Littlewood conjecture gives an asymptotic formula for the number of primes of the form $ax^2+bx+c$. We establish a relationship between the Hardy-Littlewood conjecture and the Mazur conjecture.

Let $D\in\Bbb Z$ be an integer which is neither a square nor a cube in $\Bbb Q(\sqrt {-3}),$ and let $E_D$ be the elliptic curve defined by $y^2=x^3+D.$  Mazur conjectured that the number of anomalous primes less then $N$ should be given asymptotically by $c\sqrt{N}/$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 $D=80d^6$, where $0\neq d\in \Bbb Z[\frac {1+\sqrt {-3}}{2}]$ with $d^6\in \Bbb Z.$

Conversely, if the Mazur conjecture holds for some $D$, then the polynomial $12x^2+18x+7$ represents infinitely many primes.

The main results of my talk have appeared in Proc. London Math. Soc. (3) 112 (2016) 415-453.

See also http://plms.oxfordjournals.org/content/112/2/415.full.pdf+html