Lucas numbers which are concatenations of two repdigits (Q2024020)

Let \(\{L_n\}_{n\ge 0}\) be the Lucas-companion of the Fibonacci sequence given by \(L_0=2,~L_1=1\) and \(L_{n+2}=L_{n+1}+L_n\) for all \(n\ge 0\). In the paper under review, the authors show that the only Lucas numbers \(L_n\) which are concatenations of two rep-digits; that is, their base \(10\)-representation is of the form \(a\cdots ab\cdots b\), where \(a,b\in \{0,\ldots,9\}\), \(a\ne 0\), are \(11,18,29,47,76,199,322\). The proofs use linear forms in logarithms of algebraic numbers to show that \(n<4.43\cdot 10^{29}\), and computations with continued fractions to reduce the above bound to a small value (in this case \(n\le 180\)), which allows all solutions to be found by simply listing all base \(10\)-representations of the Lucas numbers in the remaining small range.