Repdigits in base b as product of two k-generalized Pell numbers

Abstract

Let \(k\ge 2\) be an integer. The k-generalized Pell sequence \((P_{n}^{(k)})_{n\ge 2-k}\) is defined by the initial values \(0,0,\ldots ,0,1\)(k terms) and the recurrence \(P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\ldots +P_{n-k}^{(k)}\) for all \(n\ge 2\). In this study, we deal with the Diophantine equation

$$P_{n}^{(k)}P_{m}^{(k)}=d\left( \frac{b^{l}-1}{b-1}\right)$$

in positive integers n, m, k, b, d, l with \(k\ge 3,l\ge 2,~2\le m\le n,\) \(2\le b\le 10,\) and \(1\le d\le b-1,\) and we show that all solutions of this equation are given by

$$\begin{aligned} P_{2}^{(k)}P_{2}^{(k)}&=(11)_{3},~P_{3}^{(k)}P_{2}^{(k)}=(22)_{4}=(11)_{9}\text {, }P_{4}^{(k)}P_{2}^{(k)}=(222)_{3}\text { for }k\ge 3,\\ P_{5}^{(k)}P_{3}^{(k)}&=(2222)_{4}\text { for }k\ge 4, \end{aligned}$$

and

$$P_{5}^{(3)}P_{2}^{(3)}=\left( 66\right) _{10}.$$