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.