The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their index

Abstract

Let $\mathbf{D}=(D_n{)}_{n\ge 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define ${\mathcal{A}}_{E,k}=\{n\ge 1:\gcd (n,D_n)=k\}$. We give an explicit structural description of ${\mathcal{A}}_{E,k}$. Also, we explain when ${\mathcal{A}}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of ${\mathcal{A}}_{E,k}$ using the Möbius function.

15 pages

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