  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.
