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 $P_4 = 12 = 1\|2$.