Diophantine equations for second-order recursive sequences of polynomials (Q2730923)
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scientific article; zbMATH DE number 1625247
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Diophantine equations for second-order recursive sequences of polynomials |
scientific article; zbMATH DE number 1625247 |
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29 July 2001
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higher-order Diophantine equation
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recursive sequence
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Dickson polynomial
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generalized Fibonacci polynomials
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0.8943031
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0.8912647
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0.8850228
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0.8816273
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Diophantine equations for second-order recursive sequences of polynomials (English)
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Let \(G_n(X)\) be the sequence of generalized Fibonacci polynomials defined by NEWLINE\[NEWLINEG_0(X)=0, \quad G_1(X)=1, \quad G_{n+1}(X)= xG_n(X)+ BG_{n-1}(X), \quad n\in \mathbb{N},NEWLINE\]NEWLINE where \(B\) is a nonzero integer. In this paper the authors, using a finiteness criterion [\textit{Y. Bilu} and \textit{R. Tichy}, Acta Arith. 95, 261-288 (2000; Zbl 0958.11049)], prove that the equation \(G_n(X)= G_m(Y)\), where \(m,n\geq 3\) and \(m\neq n\), has only finitely many integer solutions. Furthermore, some effective results for the cases \(n=3,5\) are given.
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