  We present a rigorous study of momentum distribution and p-wave contacts of
one dimensional (1D) spinless Fermi gases with an attractive p-wave
interaction. Using the Bethe wave function, we analytically calculate the
large-momentum tail of momentum distribution of the model. We show that the
leading ($\sim 1/p^{2}$) and sub-leading terms ($\sim 1/p^{4}$) of the
large-momentum tail are determined by two contacts $C_2$ and $C_4$, which we
show, by explicit calculation, are related to the short-distance behaviour of
the two-body correlation function and its derivatives. We show as one increases
the 1D scattering length, the contact $C_2$ increases monotonically from zero
while $C_4$ exhibits a peak for finite scattering length. In addition, we
obtain analytic expressions for p-wave contacts at finite temperature from the
thermodynamic Bethe ansatz equations in both weakly and strongly attractive
regimes.
