Repdigits in generalized Pell sequences. (English) Zbl 07285963

Summary: For an integer \(k\ge 2\), let \((P_n^{(k)})_n\) be the \(k\)-generalized Pell sequence which starts with \(0,\ldots ,0,1\) (\(k\) terms) and each term afterwards is given by the linear recurrence \(P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\). In this paper, we find all \(k\)-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence \((P_n^{(2)})_n\).


11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11J86 Linear forms in logarithms; Baker’s method
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