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THE QUOTIENT SET OF
$k$-GENERALISED FIBONACCI NUMBERS IS DENSE IN 
$\mathbb{Q}_{p}$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 09 January 2017, pp. 24-29
- Print publication:
- August 2017
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- Article
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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 
$p$.
