  The ground state of two-dimensional electron systems (2DESs) at low Landau
level filling factors ($\nu\lesssim1/6$) has long been a topic of interest and
controversy in condensed matter. Following the recent breakthrough in the
quality of ultra-high-mobility GaAs 2DESs, we revisit this problem
experimentally and investigate the impact of reduced disorder. In a GaAs 2DES
sample with density $n=6.1\times10^{10}$ /cm$^2$ and mobility
$\mu=25\times10^6$ cm$^2$/Vs, we find a deep minimum in the longitudinal
magnetoresistance ($R_{xx}$) at $\nu=1/7$ when $T\simeq104$ mK. There is also a
clear sign of a developing minimum in the $R_{xx}$ at $\nu=2/13$. While
insulating phases are still predominant when $\nu\lesssim1/6$, these minima
strongly suggest the existence of fractional quantum Hall states at filling
factors that comply with the Jain sequence $\nu=p/(2mp\pm1)$ even in the very
low Landau level filling limit. The magnetic field dependent activation
energies deduced from the relation $R_{xx}\propto e^{E_A/2kT}$ corroborate this
view, and imply the presence of pinned Wigner solid states when $\nu\neq
p/(2mp\pm1)$. Similar results are seen in another sample with a lower density,
further generalizing our observations.
