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

Abstract: 

It is open whether there exists a polynomial in one variable of degree >1>1  that represents infinitely many primes.  For example, at present, we do not know whether the polynomial x2+1x2+1 represents infinitely many primes. The Hardy-Littlewood conjecture gives an asymptotic formula for the number of primes of the form ax2+bx+cax2+bx+c. We establish a relationship between the Hardy-Littlewood conjecture and the Mazur conjecture.

Let D∈ZD∈Z be an integer which is neither a square nor a cube in Q(−3−−−√),Q(−3), and let EDED be the elliptic curve defined by y2=x3+D.y2=x3+D.  Mazur conjectured that the number of anomalous primes less then NN should be given asymptotically by cN−−√/cN/logNN(cc is a positive constant), and in particular there should be infinitely many anomalous primes for EDED.  We show that the Hardy-Littlewood conjecture implies the Mazur conjecture, except for D=80d6D=80d6, where 0≠d∈Z[1+−3√2]0≠d∈Z[1+−32] with d6∈Z.d6∈Z.

Conversely, if the Mazur conjecture holds for some DD, then the polynomial 12x2+18x+712x2+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