Pell and Pell–Lucas Numbers as Product of Two Repdigits

Abstract

In this study, we find all Pell and Pell–Lucas numbers that are product of two repdigits in the base $b$ for $b\in[2,10]$. It is shown that the largest Pell and Pell–Lucas numbers that can be expressed as a product of two repdigits are $P_{7}=169$ and $Q_{6}=198$, respectively. Also, we have the representations

$$P_{7}=169=(111)_{3}\times(111)_{3}$$

and

$$Q_{6}=198=2\times99=3\times66=6\times33=9\times22.$$

Furthermore, it is shown in the paper that the equation $P_{k}=(b^{n}-1)(b^{m}-1)$ has only the solution $(b,k,m,n)=(2,1,1,1)$ and the equation $Q_{k}=(b^{n}-1)(b^{m}-1)$ has no solution $(b,k,m,n)$ in positive integers for $2\leq$ $b\leq10$. The proofs depend on lower bounds for linear forms and some tools from Diophantine approximation.