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

Fundamenta Mathematicae (1998)

• Volume: 156, Issue: 2, page 111-129
• ISSN: 0016-2736

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## Abstract

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Let Sq denote the set of squares, and let $S{Q}_{n}$ be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let ${B}_{n}\left(x,y\right)=\left(\genfrac{}{}{0pt}{}{x+y}{x}\right)MODn$. For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.

## How to cite

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Bès, Alexis, and Korec, Ivan. "Definability within structures related to Pascal’s triangle modulo an integer." Fundamenta Mathematicae 156.2 (1998): 111-129. <http://eudml.org/doc/212264>.

@article{Bès1998,
abstract = {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; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.},
author = {Bès, Alexis, Korec, Ivan},
journal = {Fundamenta Mathematicae},
keywords = {Pascal's triangle modulo n; decidability; definability; decidability of theories; squaring function; Pascal triangle},
language = {eng},
number = {2},
pages = {111-129},
title = {Definability within structures related to Pascal’s triangle modulo an integer},
url = {http://eudml.org/doc/212264},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Bès, Alexis
AU - Korec, Ivan
TI - Definability within structures related to Pascal’s triangle modulo an integer
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 2
SP - 111
EP - 129
AB - 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; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.
LA - eng
KW - Pascal's triangle modulo n; decidability; definability; decidability of theories; squaring function; Pascal triangle
UR - http://eudml.org/doc/212264
ER -

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