  We present a phase-coherent timing solution for PSR J1640-4631, a young 206
ms pulsar using X-ray timing observations taken with NuSTAR. Over this timing
campaign, we have measured the braking index of PSR J1640-4631 to be n =
3.15+/-0.03. Using a series of simulations, we argue that this unusually high
braking index is not due to timing noise, but is intrinsic to the pulsar's
spin-down. We cannot, however, rule out contamination due to an unseen glitch
recovery, although the recovery timescale would have to be longer than most yet
observed. If this braking index is eventually proven to be stable, it
demonstrates that pulsar braking indices greater than 3 are allowed in nature,
hence other physical mechanisms such as mass or magnetic quadrupoles are
important in pulsar spin-down. We also present a 3-sigma upper limit on the
pulsed flux at 1.4 GHz of 0.018 mJy.
