MATHEMATICA BOHEMICA, Vol. 137, No. 4, pp. 403-413, 2012

On the intersection of two distinct $k$-generalized Fibonacci sequences

Diego Marques

Diego Marques, University of Brasilia, CEP: 70910-900, Brasilia, Brazil, e-mail:

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 $F\cap T=\{0,1,2,13\}$.

Keywords: $k$-generalized Fibonacci numbers, linear forms in logarithms, reduction method

Classification (MSC 2010): 11B39, 11D61, 11J86

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