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

Let k≥ 2 k ≥ 2 and ₙ⁽ᵏ⁾≥ 2-k { 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.