Vol. 263, No. 2, 2013

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Elliptic aliquot cycles of fixed length

Nathan Jones

Vol. 263 (2013), No. 2, 353–371
Abstract

Silverman and Stange define the notion of an aliquot cycle of length L for a fixed elliptic curve E over , 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.

Keywords
elliptic curve, aliquot cycle, amicable pair
Mathematical Subject Classification 2010
Primary: 11G05
Milestones
Received: 16 July 2012
Revised: 10 December 2012
Accepted: 14 December 2012
Published: 31 May 2013
Authors
Nathan Jones
Department of Mathematics
University of Mississippi
Hume Hall 305
P.O. Box 1848
University, MS 38677
United States