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Zbl 0773.11077
Sun, Zhiwei
Reduction of unknowns in diophantine representations. (English)
[J] Sci. China, Ser. A 35, No.3, 257-269 (1992). ISSN 1006-9283; ISSN 1862-2763/e

Authors' summary: The hardest step to solve Hilbert's tenth problem is to prove that the exponential relation is diophantine. In the study of decision problems concerning the solvability of diophantine equations with few unknowns, reducing unknowns in diophantine representations plays an important role. In this paper, we give diophantine representations of $C=\psi\sb B(A,1)$ (where $\psi\sb 0(A,1)=0$, $\psi\sb 1(A,1)=1$, $\psi\sb{m+1}(A,1)=A\psi\sb m(A,1)-\psi\sb{m-1}(A,1))$ and $W=V\sp B \wedge A\sb 1,\dots,A\sb k\in\square\wedge$ $S \vert T \wedge R > 0$ with only 3 and 5 natural number unknowns respectively, $C= \psi\sb B(A,1)$ (on the condition $1< \vert B \vert <{\vert A \vert \over 2}-1)$ and $W=V\sp B \wedge A\sb 1, \dots, A\sb k \in \square \wedge S \vert T$ with 4 and 6 integer unknowns respectively.
[T.Pheidas (Iraklion)]

MSC 2000:: *11U05 Decidability related to number theory
03B25 Decidability of theories and sets of sentences
11U09 Connections of number theory with model theory
03C60 Model-theoretic algebra

Keywords: undecidability; Lucas sequence; Hilbert's tenth problem; diophantine representations

Cited in: Zbl 0793.03004

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