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The sign of an elliptic divisibility sequence

Joseph H. Silverman·Nelson Stephens
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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.

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Joseph H. Silverman, Nelson Stephens. 2004-02-25. The sign of an elliptic divisibility sequence. https://arxiv.org/abs/math/0402415

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