On the Diophantine equation \(G_n(x)=G_m(P(x))\) for third order linear recurring sequences
Let \(\mathbf{K}\) be a field of characteristic \(0\) and let \(a, b, c, G_{0}, G_{1}, G_{2}, P \in \mathbf{K}[x],\) \(\deg P \geq 1\). Further let the sequence of polynomials \((G_{n}(x))^{\infty}_{n=0}\) be defined by the third order linear recurring sequence \(G_{n+3}(x)= a(x) G_{n+2}(x) + b(x) G_{n+1}(x)+ c(x) G_{n}(x)\) for \(n \geq 0\). In this paper the author gives conditions under which the Diophantine equation \[ G_{n}(x)= G_{m}(P(x)) \] has at most \(\exp (10^{24})\) many solutions \((n, m) \in \mathbb{Z}^2\), \(n, m \geq 0\). The proof uses a result on \(S\)-unit equations over fields of characteristic \(0\) due to \textit{J.-H. Evertse, H. P. Schlickewei} and \textit{W. M. Schmidt} [Ann. Math. (2) 155, No. 3, 807--836 (2002; Zbl 1026.11038)]. This paper is a continuation of the joint work of the author with \textit{A. Pethő} and \textit{R. F. Tichy} on this equation in the case of second order linear recurring sequences [Monatsh. Math. 137, No. 3, 173--196 (2002; Zbl 1026.11039)].
scientific article; zbMATH DE number 2067403
scientific article