  We derive the theoretical limit of single-photon purity of heralded
single-photon sources, and accordingly demonstrate a bright, gigahertz-pulsed
heralded source with the purity saturating the limit. Based on spontaneous
four-wave mixing in a silicon spiral waveguide, this on-chip source is measured
to have a coincidence rate exceeding 1.5 MHz at a coincidence to accidental
(CAR) ratio of 16.77. The single-photon purity, quantified by the
auto-correlation function $g^{(2)}_h(0)$, reaches the theoretical limit with
the lowest value of $0.00094 \pm 0.00002$ obtained at a coincidence rate of 0.8
kHz. We attribute our results to effective spectral filtering as well as the
coherent pump condition helped by optical injection locking.
