On the Diophantine equation $G_{n}(x)=G_{m}(P(x))$ for third order linear recurring sequences

Clemens Fuchs

Abstract: Let ${\bf K}$ be a field of characteristic $0$ and let $a,b,c,G_{0},G_{1},G_{2},P\in{\bf K}[x]$, $\deg P\geq 1$. Further let the sequence of polynomials $(G_{n}(x))_{n=0}^{\infty}$ be defined by the third order linear recurring sequence
$$ G_{n+3}(x)=a(x)G_{n+2}(x)+b(x)G_{n+1}(x)+c(x)G_{n}(x)\rmqd{for} n\geq 0. $$
In this paper we give conditions under which the Diophantine equation
$$ G_{n}(x)=G_{m}(P(x)) $$
has at most $\exp(10^{24})$ many solutions $(n,m)\in \Z^{2}$, $n,m\geq 0$. The proof uses a very recent result on $S$-unit equations over fields of characteristic $0$ due to J.-H. Evertse, H.P. Schlickewei and W.M. Schmidt (cf. [8]). This paper is a continuation of the joint work of the author with A. Petho and R.F. Tichy on this equation in the case of second order linear recurring sequences (cf. [9]).

Keywords: Diophantine equations; ternary linear recurring sequences; $S$-unit equations.

Classification (MSC2000): 11D45, 11D04, 11D61, 11B37.

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