The Density of the Terms in an Elliptic Divisibility Sequence Having a Fixed G.c.d. With Their Index
Year: 2017
Type: preprint
Abstract: Let $\mathbf{D}=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of $\mathscr{A}_{E,k}$. Also, we explain when $\mathscr{A}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of $\mathscr{A}_{E,k}$ using the Möbius function.
Source: arXiv (Cornell University)
Author: Seoyoung Kim
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6959840480