  Let $K$ be a number field and let $E$ be an elliptic curve defined and of
rank one over $K$. For a set $\calW_K$ of primes of $K$, let
$O_{K,\calW_K}=\{x\in K: \ord_{\pp}x \geq 0, \forall \pp \not \in \calW_K\}$.
Let $P \in E(K)$ be a generator of $E(K)$ modulo the torsion subgroup. Let
$(x_n(P),y_n(P))$ be the affine coordinates of $[n]P$ with respect to a fixed
Weierstrass equation of $E$. We show that there exists a set $\calW_K$ of
primes of $K$ of natural density one such that in $O_{K,\calW_K}$
multiplication of indices (with respect to some fixed multiple of $P$) is
existentially definable and therefore these indices can be used to construct a
Diophantine model of $\Z$. We also show that $\Z$ is definable over
$O_{K,\calW_K}$ using just one universal quantifier. Both, the construction of
a Diophantine model using the indices and the first-order definition of $\Z$
can be lifted to the integral closure of $O_{K,\calW_K}$ in any infinite
extension $K_{\infty}$ of $K$ as long as $E(K_{\infty})$ is finitely generated
and of rank one.
