The sign of an elliptic divisibility sequence

TL;DR

This paper derives a formula for the sign of terms in elliptic divisibility sequences and explores their properties, including implications for dynamical systems and sequence realizability.

Contribution

It provides a new explicit formula for the sign of elliptic divisibility sequence terms and investigates their dynamical system realizability.

Findings

Sign(W_n) = (-1)^[n*b] for irrational b in typical cases

The absolute value sequence cannot be realized as fixed points of any dynamical system

The paper connects elliptic divisibility sequences with dynamical systems theory

Abstract

An elliptic divisibility sequence (EDS) is a sequence of integers W_0,W_1,W_2,... generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of W_n for unbounded nonsingular elliptic divisibility sequences. A typical case is Sign(W_n) = (-1)^[n*b] for an irrational real number b, where [x] denotes the greatest integer in x. As an application, we show that the associated sequence of absolute values |W_1|,|W_2|,|W_3|,... cannot be realized as the sequence counting fixed points of any (abstract) dynamical system.