On concatenations of two $k$-generalized Pell numbers

Abstract

We study the concatenation of two $k$-generalized Pell numbers. More precisely, we determine all solutions of the equation $P_n^{(k)}=P_m^{(k)}\cdot 10^d+P_p^{(k)}$, where $d$ is the number of decimal digits of $P_p^{(k)}$. We prove that for $k\ge 3$ there are no solutions, while for $k=2$ the only solution is $P_4=12=1\Vert 2$.