The sign of an elliptic divisibility sequence
Joseph H. Silverman, Nelson Stephens
Abstract
An elliptic divisibility sequence (EDS) is a sequence of integers W0,W1,W2,... generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of Wn for unbounded nonsingular elliptic divisibility sequences. A typical case is Sign(Wn) = (-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 |W1|,|W2|,|W3|,... cannot be realized as the sequence counting fixed points of any (abstract) dynamical system.
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