Journal of the Ramanujan Mathematical Society

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The Sign of an Elliptic Divisibility Sequence


Affiliations
1 Mathematics Department, Box 1917, Brown University, Providence, RI 02912, United States
2 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom
     

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An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 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 EDS, a typical case being Sign (Wn)=(-1) └nβ┘ for an irrational number β∈ℝ. As an application, we show that the associated sequence of absolute values (|Wn|) cannot be realized as the fixed point counting sequence of any abstract dynamical system.
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  • The Sign of an Elliptic Divisibility Sequence

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Authors

Joseph H. Silverman
Mathematics Department, Box 1917, Brown University, Providence, RI 02912, United States
Nelson Stephens
Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom

Abstract


An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 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 EDS, a typical case being Sign (Wn)=(-1) └nβ┘ for an irrational number β∈ℝ. As an application, we show that the associated sequence of absolute values (|Wn|) cannot be realized as the fixed point counting sequence of any abstract dynamical system.