Article contents
THE QUOTIENT SET OF
$k$-GENERALISED FIBONACCI NUMBERS IS DENSE IN
$\mathbb{Q}_{p}$
Published online by Cambridge University Press: 09 January 2017
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
$p$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2017 Australian Mathematical Publishing Association Inc.
References
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