Repdigits as sums of two $$k$$ k -Fibonacci numbers

Abstract

For an integer \(k\ge 2\), let \((F_{n}^{(k)})_{n}\) be the \(k\)-Fibonacci sequence which starts with \(0,\ldots ,0,1\) (\(k\) terms) and each term afterwards is the sum of the \(k\) preceding terms. In this paper, we find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion) which are sums of two \(k\)-Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker–Davenport reduction method. This paper is an extended work related to our previous work (Bravo and Luca Publ Math Debr 82:623–639, 2013).