THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN

Abstract

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$ . Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$ , Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly , to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$ -adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$ . For any integer $k\geq 2$ , let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$ -generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ( $k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$ -adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$ -generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$ .