Year: 2026
Type: preprint
Abstract: We study multiplicative dependence between terms of the \(k\)-generalized Pell sequence \((P_n^{(k)})_{n\ge 2-k}\), defined by the linear recurrence \[ P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}, \] with initial conditions \(P_0^{(k)}=\cdots=P_{-(k-2)}^{(k)}=0\) and \(P_1^{(k)}=1\). We prove that for every \(k\ge 2\) and all integers \(n>m\ge 2\), the equation \[ \bigl(P_n^{(k)}\bigr)^x=\bigl(P_m^{(k)}\bigr)^y \] has no solution in positive integers \(x,y\) with \(x
Source: arXiv (Cornell University)
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161711450