  We have investigated the magnetism and pairing correlations of the triangular
lattice based on the Hubbard model using the determinant quantum Monte Carlo
method and the constrained path Monte Carlo. The results show that the presence
of the next-nearest-neighbor hopping integral $t^{\prime}$ introduces an
additional energy scale to the system, and through $t^{\prime}$, one can
regulate the shape of the density of states and thus the position of the van
Hove singularity point. Increasing inverse temperature $\beta$ and on-site
interaction $U$ favor the formation of ferromagnetic correlation in a rather
large filling region, and the calculations for different lattice sizes show
that the range of the ferromagnetic correlations is smaller than the smallest
lattice simulated at the investigated temperatures. We study the different
pairing correlations of the triangular lattice near several typical fillings
and show that the $f$-wave pairing dominates the system in the filling region
near the van Hove singularity point with a high density of states, where the
ferromagnetic correlation is also enhanced. When the filling is close to
half-filling, the pairing susceptibility with $f$ wave is suppressed and the
pairing susceptibility of $f_n$ wave is enhanced, however, both the effective
pairing interaction with $f$ wave and $f_n$ wave are negative, which indicates
that neither $f$-wave nor $f_n$-wave superconductivity may exist. Finally, we
find that the pairing channel of different symmetry in the system maybe closely
related to the magnetic properties. Ferromagnetic fluctuation favors the
formation of $f$-wave pairing, while antiferromagnetic fluctuation tends to
promote $f_n$-wave pairing.
