## Number Theory Seminar

### Thursday, May 10th, 2018

**Time:** 2:30-3:20p.m. **Place:** Jeffery Hall 422

**Speaker:** Seoyoung Kim (Brown University)

**Title:** The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their indices.

**Abstract:** Let $\D=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\A_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of $\A_{E,k}$. Also, we explain when $\A_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, with preconditions, we obtain an explicit density for $\A_{E,k}$ using the M\"obius function. The precondition holds when $E$ is a finitely anomalous elliptic curve.