On concatenations of two $k$-generalized Lucas numbers

Abstract: For an integer ( k \geq 2 ), the sequence of ( k )-generalized Lucas numbers is defined by the recurrence relation ( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} ) for all ( n \geq 2 ), with initial conditions ( L_0^{(k)} = 2 ), ( L_1^{(k)} = 1 ) for all ( k \geq 2 ), and ( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 ) for ( k \geq 3 ). In this paper, we determine all ( k )-generalized Lucas numbers that are concatenations of two terms of the same sequence and completely solve this problem for ( k \geq 3 ). Our approach combines nonzero lower bounds for linear forms in logarithms, reduction techniques based on the Baker--Davenport method and the LLL-algorithm, together with continued fraction analysis and computational verification using SageMath.