k-generalized Fibonacci numbers that are palindromic concatenations of two distinct repdigits

Abstract Let k 2 k ≥ 2 and \Fₙ^{ (k) \}₍ ₂-₊ F n (k) n ≥ 2 - k be the sequence of k -generalized Fibonacci numbers whose first k terms are 0, , 0, 0, 1 0, …, 0, 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we determine all terms of this sequence that are palindromic concatenations of two distinct repdigits. We show that F₁₁^ (5) =464 F 11 (5) = 464 is the only such term. Our proof transitionally employs Matveev’s theorem for lower bounds on linear forms in logarithms and the LLL-algorithm to reduce the large initial bounds on the variables. For large k, we utilize the fact that k -generalized Fibonacci numbers are very close to powers of two.