Let k ≥ 3 be an odd integer. Consider the k-generalized Fibonacci sequence backward. The characteristic polynomial of this sequence has no dominating zero, therefore the application of Baker's method becomes more difficult. In this paper, we investigate the coincidence of the absolute values of two terms. The principal theorem gives a lower bound for the difference of two terms (in absolute value) if the larger subscript of the two terms is large enough. A corollary of this theorem makes possible to bound the coincidences in the sequence. The proof essentially depends on the structure of the zeros of the characteristic polynomial, and on the application of linear forms in the logarithms of algebraic numbers. Then we reduced the theoretical bound in practice for 3 ≤ k ≤ 99, and determined all the coincidences in the corresponding sequences. Finally, we explain certain patterns of pairwise occurrences in each sequence depending on k if k exceeds a suitable entry value associated to the pair.

The authors are grateful to A. Mehdaoui, Sz. Tengely, T. Wurth, T. Bartalos, and Gy. Bugar for their kind help in carrying out the computations. F. Luca worked on this paper while he visited Max Planck Institute for Software Systems in Saarbrucken, Germany in the Fall of 2020. He thanks this Institution for hospitality and support. For L. Szalay the research and this work was supported by Hungarian National Foundation for Scientific Research Grant No. 128088, and No. 130909, and by the Slovak Scientific Grant Agency VEGA 1/0776/21.