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

TL;DR

This paper proves a conjecture about the unique solutions to a Diophantine equation involving intersections of different k-generalized Fibonacci sequences, confirming that only two specific solutions exist.

Contribution

The paper confirms a longstanding conjecture that only two solutions exist for the intersection of k-generalized Fibonacci sequences under given conditions.

Findings

Confirmed the conjecture with only two solutions.

Established the uniqueness of solutions for the intersection problem.

Validated the specific solutions (7,6,3,2) and (12,11,7,3).

Abstract

For $k\geq 2$, 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 $F_m^{(k)}=F_n^{(\ell)}$, with $\ell>k>1, n>\ell+1, m>k+1$ are [(m,n,\ell,k)=(7,6,3,2) and (12,11,7,3).] In this paper, we confirm this conjecture.