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
$p$
.