A framework for deterministic primality proving using elliptic curves   with complex multiplication
  We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes, including the largest prime currently known whose primality cannot feasibly be proved using classical methods. 
Alexander Abatzoglou, Alice Silverberg, Andrew V. Sutherland, and   Angela Wong
math.NT