The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

摘要

For k ≥ 2, the k-generalized Fibonacci sequence (F n (k) ) n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F m (k) = F n (ℓ) , with ℓ > k > 1, n > ℓ + 1, m > k + 1 are $$(m,n,ell ,k) = (7,6,3,2)and(12,11,7,3)$$. In this paper, we confirm this conjecture. Keywords k-generalized Fibonacci numbers linear forms in logarithms intersection Mathematical subject classification 11B39 11J86 Supported by FAP-DF, FEMAT and CNPq-Brazil.