  Let $Q$ be the 3-Kronecker quiver, i.e., $Q$ has two vertices, labeled by 1
and 2, and three arrows from 2 to 1. Fix an algebraically closed field $k$. Let
$\mathcal{C}$ be a regular component of the Auslander-Reiten quiver containing
an indecomposable module $X$ with dimension $(1,1)$ or $(2,1)$. Using the
properties of the Fibonacci numbers, we show that the Gabriel-Roiter measures
of the indecomposable modules in $\mathcal{C}$ are uniquely determined by the
dimension vectors. In other words, two indecomposable modules in $\mathcal{C}$
are not isomorphic if and only if their Gabriel-Roiter measures are different.
