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THE QUOTIENT SET OF $k$-GENERALISED FIBONACCI NUMBERS IS DENSE IN $\mathbb{Q}_{p}$

Part of: Sequences and sets

Published online by Cambridge University Press:  09 January 2017

CARLO SANNA
CARLO SANNA*
Affiliation:
Department of Mathematics, Università degli Studi di Torino, Turin, Italy email [email protected]
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Abstract

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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 $p$.

Keywords

Fibonacci numbersk-generalised Fibonacci numbersp-adic numbersdensity

MSC classification

Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Secondary: 11B37: Recurrences 11B05: Density, gaps, topology
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 24 - 29
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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