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  1. Alexis Bés & Denis Richard (1998). Undecidable Extensions of Skolem Arithmetic. Journal of Symbolic Logic 63 (2):379-401.
    Let $ be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of $\langle\mathbb{N};\bot, , is undecidable. Now denote by $ the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure $\langle\mathbb{N};\bot, . Furthermore, the structures $\langle\mathbb{N};\mid, and $\langle\mathbb{N};=,+,x\rangle$ are interdefinable.
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  2. Patrick Cegielski, Leszek Pacholski, Denis Richard, Jerzy Tomasik & Alex Wilkie (1997). Preface. Annals of Pure and Applied Logic 89 (1):1.
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  3. Patrick Cegielski, Yuri Matiyasevich & Denis Richard (1996). Definability and Decidability Issues in Extensions of the Integers with the Divisibility Predicate. Journal of Symbolic Logic 61 (2):515-540.
    Let M be a first-order structure; we denote by DEF(M) the set of all first-order definable relations and functions within M. Let π be any one-to-one function from N into the set of prime integers. Let ∣ and $\bullet$ be respectively the divisibility relation and multiplication as function. We show that the sets DEF(N,π,∣) and $\mathrm{DEF}(\mathbb{N},\pi,\bullet)$ are equal. However there exists function π such that the set DEF(N,π,∣), or, equivalently, $\mathrm{DEF}(\mathbb{N},\pi,\bullet)$ is not equal to $\mathrm{DEF}(\mathbb{N},+,\bullet)$ . Nevertheless, in all cases (...)
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  4. Denis Richard (1989). Definability in Terms of the Successor Function and the Coprimeness Predicate in the Set of Arbitrary Integers. Journal of Symbolic Logic 54 (4):1253-1287.
    Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate $\perp$ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are $\{S, \perp\}$ -definable over Z. Applications to definability over Z and N (...)
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  5. Denis Richard (1985). Answer to a Problem Raised by J. Robinson: The Arithmetic of Positive or Negative Integers is Definable From Successor and Divisibility. Journal of Symbolic Logic 50 (4):927-935.
    In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]).
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