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Preprint Number 170
170. Alexandra Shlapentokh 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 Submission date: 5 February 2009. Abstract: 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_{p}x \geq 0, \forall p \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 Mathematics Subject Classification: 11U05, 11G05 Keywords and phrases: |

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