Narayana numbers that are products of two Fibonacci numbers

TL;DR

This paper explicitly solves the Diophantine equation relating Narayana numbers to products of two Fibonacci numbers, identifying all such Narayana numbers that are Fibonacci products.

Contribution

It provides a complete characterization of Narayana numbers that can be expressed as the product of two Fibonacci numbers, solving an open problem.

Findings

Identifies all Narayana numbers that are products of two Fibonacci numbers.

Provides explicit solutions to the Diophantine equation involving Narayana and Fibonacci sequences.

Establishes a complete list of such special Narayana numbers.

Abstract

Let $\{N_m\}_{m\ge0}$ be the Narayana's cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and \[ N_{m+3}=N_{m+2}+N_m,\quad \text{ for }\; m\geq 0 \] and let $\{F_n\}_{n\ge0}$ be the Fibonacci sequence. In this paper we solve explicitely the Diophantine equation \[ N_m=F_nF_k, \] in positive unknowns $m,\,n$ and $k$. That is, we find the non-zero narayana numbers that are products of two Fibonacci numbers.