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Definability within structures related to Pascal’s triangle modulo an integer

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Let Sq denote the set of squares, and let $SQ_n$ be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let $B_n(x,y)=({x+y \atop x}) MOD n$. For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; B_n,⊥⟩ and ⟨ℕ; B_n,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; B_p,SQ_p⟩ is decidable.
  • Université Paris 7, Equipe de Logique URA 753, 2, place Jussieu, 75251 Paris Cedex 05, France
  • Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 81473 Bratislava, Slovakia
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