| Abstrakt: |
Abstract: Let k\geq2 and define F^{(k)}:=(F_n^{(k)})_{n\geq0}, the k-generalized Fibonacci sequence whose terms satisfy the recurrence relation F_n^{(k)}=F_{n-1}^{(k)}+F_{n-2}^{(k)}+\cdots+ F_{n-k}^{(k)}, with initial conditions 0,0,\dots,0,1 (k terms) and such that the first nonzero term is F_1^{(k)}=1. The sequences F:=F^{(2)} and T:=F^{(3)} are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation F_n^{(k)}=F_m^{(\ell)}. In this note, we use transcendental tools to provide a general method for finding the intersections F^{(k)}\cap F^{(m)} which gives evidence supporting the Noe-Post conjecture. In particular, we prove that F\cap T=\{0,1,2,13\}. |
