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Sequences generated by elliptic curves

Volume 188 / 2019

Betül Gezer, Osman Bizim Acta Arithmetica 188 (2019), 253-268 MSC: Primary 14H52; Secondary 11G07, 14G20, 11B37. DOI: 10.4064/aa170504-25-6 Published online: 7 March 2019

Abstract

We study the properties of the sequences $(G_{n}(P))_{n\geq 0} $ and $(H_{n}(P))_{n\geq 0}$ generated by the numerators of the $x$- and $y$-coordinates of the multiples of a point $P$ on an elliptic curve $% E$ defined over a field $K$. We prove that if $E$ is defined over a finite field, then these sequences are purely periodic. Then we generalize this result to the case of modulo prime powers. As a consequence, we deduce that certain subsequences of these sequences converge $p$-adically, i.e., are $\mathbb{Z}_{p}$-Cauchy.

Authors

  • Betül GezerDepartment of Mathematics
    Faculty of Science
    Bursa Uludağ University
    Görükle, 16059, Bursa, Turkey
    e-mail
  • Osman BizimDepartment of Mathematics
    Faculty of Science
    Bursa Uludağ University
    Görükle, 16059, Bursa, Turkey
    e-mail

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