Fast Radio Bursts are millisecond-duration astronomical radio pulses of unknown physical origin that appear to come from extragalactic distances. Previous follow-up observations have failed to find additional bursts at the same dispersion measures (i.e. integrated column density of free electrons between source and telescope) and sky position as the original detections. The apparent non-repeating nature of the fast radio bursts has led several authors to hypothesise that they originate in cataclysmic astrophysical events. Here we report the detection of ten additional bursts from the direction of FRB121102, using the 305-m Arecibo telescope. These new bursts have dispersion measures and sky positions consistent with the original burst. This unambiguously identifies FRB121102 as repeating and demonstrates that its source survives the energetic events that cause the bursts. Additionally, the bursts from FRB121102 show a wide range of spectral shapes that appear to be predominantly intrinsic to the source and which vary on timescales of minutes or shorter. While there may be multiple physical origins for the population of fast radio bursts, the repeat bursts with high dispersion measure and variable spectra specifically seen from FRB121102 support models that propose an origin in a young, highly magnetised, extragalactic neutron star.
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