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On polynomial values of the sum and the product of the terms of linear recurrences. (English) Zbl 0973.11034
Let $$x\geq 2$$, $$x_1,x_2,\dots,x_m$$ be positive integer variables. The author discusses the solubility of Diophantine equations of the form $\sum^m_{i=1} G^{(i)}_{x_i}=F(x)\;\text{and} \prod^m_{i=1} G^{(i)}_{x_i}=F(x),$ where $$G^{(i)}=\{G^{(i)}_{x_i}\}^\infty_x 0$$, $$i=1,2,\dots,m$$, is a system a linear recurrences of order $$k_i$$ defined by $G^{(i)}_x=A^{(i)}_1 G^{(i)}_{x-1}+A^{(i)}_2G^{(i)}_{x-2}+\dots + A^{(i)}_{k_i}G^{(i)}_{x-k_i},\;2\leq k_i\leq x,$ where $$F(x)=dx^q+d_px^p+d_{p-1}x^{p-1}+\dots+d_0\in \mathbb Z [x]$$ has degree $$q\geq 2$$ with $$q>p$$ and where for $$j=0,1,\dots,k_i-1$$, the initial values $$G^{(i)}_j$$ and the coefficients $$A^{(i)}_{j+1}$$ are rational integers. Earlier results corresponding to special cases are discussed briefly and it is shown that, under certain conditions on the recurrences $$G^{(i)}$$, on the variables $$x$$, $$x_i$$ and on $$q$$, $$p$$, the equations have no solutions for sufficiently large, effectively computable degree $$q$$.
##### MSC:
 11D04 Linear Diophantine equations 11B37 Recurrences
##### Keywords:
linear recurrences; polynomial values
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