  If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler
metric $F$ are irrationally elliptic, there exist either exactly
$2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As
an application, we show the existence of three distinct closed geodesics on
bumpy Finsler $(S^3,F)$ if any prime closed geodesic has non-zero Morse index.
