Elliptic aliquot cycles of fixed length

TL;DR

This paper refines the conjecture on the distribution of elliptic aliquot cycles of fixed length by combining heuristics, providing a precise asymptotic formula and numerical evidence for their count.

Contribution

It introduces a refined conjecture with a specific constant for counting elliptic aliquot cycles, combining Lang-Trotter and Koblitz heuristics.

Findings

Derived a precise asymptotic formula for elliptic aliquot cycles

Provided a criterion for the positivity of the conjectural constant

Presented numerical evidence supporting the refined conjecture

Abstract

Silverman and Stange define the notion of an aliquot cycle of length L for a fixed elliptic curve E defined over the rational numbers, and conjecture an order of magnitude for the function which counts such aliquot cycles. In the present note, we combine heuristics of Lang-Trotter with those of Koblitz to refine their conjecture to a precise asymptotic formula by specifying the appropriate constant. We give a criterion for positivity of the conjectural constant, as well as some numerical evidence for our conjecture.