Elliptic Reciprocity
  The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were introduced by Silverman and Stange. Settling a matter left open by Silverman and Stange it is shown that for d=3 there are elliptic cycles of length 6. For d not equal to 3 the question of the existence of proper elliptic lists of length n over d is reduced to the the theory of prime producing quadratic polynomials. For d=163 a proper elliptic list of length 40 is exhibited. It is shown that for each d there is an upper bound on the length of a proper elliptic list over d. The final section of the paper contains heuristic arguments supporting conjectured asymptotics for the number of elliptic pairs below integer X. Finally, for d congruent to 3 modulo 8 the existence of infinitely many anomalous prime numbers is derived from Bunyakowski's Conjecture for quadratic polynomials. 
Liljana Babinkostova, Kevin M. Bombardier, Matthew M. Cole, Thomas A.   Morrell and Cory B. Scott
math.NT math.AG