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

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 ^(?) , with ? > k > 1, n > ? + 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.