The sign of an elliptic divisibility sequence

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.

17 pages

Cited by in corpus (10)

• Algebraic divisibility sequences over function fields
• The uniform primality conjecture for elliptic curves
• An elliptic sequence is not a sampled linear recurrence sequence
• The Uniform Primality Conjecture for the Twisted Fermat Cubic
• The Signs in Elliptic Nets
• p-adic properties of division polynomials and elliptic divisibility sequences
• Primitive divisors on twists of the Fermat cubic
• Sequences associated to elliptic curves
• The elliptic sieve and Brauer groups
• Descent on Elliptic Curves and Hilbert's Tenth Problem