Elliptic Reciprocity

TL;DR

This paper explores elliptic analogs of amicable pairs and cycles, establishing existence results, bounds, and conjectures related to elliptic lists over square-free integers, connecting elliptic curves with prime-producing quadratic polynomials.

Contribution

It introduces elliptic pairs, cycles, and lists, proves existence of certain cycles, provides bounds on list lengths, and links the theory to prime-producing quadratic polynomials and conjectures.

Findings

Existence of elliptic cycles of length 6 for d=3.

Construction of a proper elliptic list of length 40 for d=163.

Upper bounds on the length of elliptic lists over any d.

Abstract

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.