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Let $K$ be a number field and let $E$ be an elliptic curve defined and ofrank 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 fixedWeierstrass equation of $E$. We show that there exists a set $\calW_K$ ofprimes of $K$ of natural density one such that in $O_{K,\calW_K}$multiplication of indices (with respect to some fixed multiple of $P$) isexistentially definable and therefore these indices can be used to construct aDiophantine model of $\Z$. We also show that $\Z$ is definable over$O_{K,\calW_K}$ using just one universal quantifier. Both, the construction ofa 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 infiniteextension $K_{\infty}$ of $K$ as long as $E(K_{\infty})$ is finitely generatedand of rank one.

Using Indices of Points on an Elliptic Curve to Construct A Diophantine  Model of $\Z$ and Define $\Z$ Using One Universal Quantifier in Very Large  Subrings of Number Fields, Including $\Q$