Generalized Lucas numbers which are concatenations of two repdigits (Q2041219)

The \(k\)-generalized Lucas numbers \(\{L_n^{(k)}\}_{n\ge 2-k}\) follow the same recurrence as the \(k\)-generalized Fibonacci numbers (each term is the sum of the \(k\)-preceding ones with initial terms \((0,0,\ldots,2,1)\) (a string of length \(k\) which starts with \(k-2\) zeros). The convention for the indices to start at \(2-k\) is so that the first nonzero term in the above sequence is \(L_0^{(k)}=2\). In the paper under review the authors study which numbers of this sequence look like a concatenation of two repdigits, namely \(a\cdots ab\cdots b\) when written in base \(10\), where \(a,b\in \{0,1,\ldots,9\}\). Clearly, all numbers of at most two digits are a concatenation of two repdigits, which gives them some trivial parametric families like \(96=L_7^{(k)}\) for all \(k\ge 7\). They show that there are only finitely many nontrivial solutions (with at least three digits) the largest such being \(11888=L_{14}^{(7)}\). The proof uses linear forms in logarithms à la Baker and reduction techniques based on continued fractions such as the Baker-Davenport reduction lemma.