Repdigits as Product of Terms of k -Bonacci Sequences.

For any integer k ≥ 2 , the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F − (k − 2) (k) = ⋯ = F 0 (k) = 0 and F 1 (k) = 1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form a a ... a , with a ∈ [ 1 , 9 ] ) in the sequence (F n (k) F n (k + m)) n , for m ∈ [ 1 , 9 ] . This result generalizes a recent work of Bednařík and Trojovská (the case in which (k , m) = (2 , 1) ). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method). [ABSTRACT FROM AUTHOR]