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No robust detection of prompt electromagnetic counterparts to fast radiobursts (FRBs) has yet been obtained, in spite of several multi-wavelengthsearches carried out so far. Specifically, X/gamma-ray counterparts arepredicted by some models. We planned on searching for prompt gamma-raycounterparts in the Insight-Hard X-ray Modulation Telescope (Insight-HXMT)data, taking advantage of the unique combination of large effective area in thekeV-MeV energy range and of sub-ms time resolution. We selected 39 FRBs thatwere promptly visible from the High-Energy (HE) instrument aboard Insight-HXMT.After calculating the expected arrival times at the location of the spacecraft,we searched for a significant excess in both individual and cumulative timeprofiles over a wide range of time resolutions, from several seconds down tosub-ms scales. Using the dispersion measures in excess of the Galactic terms,we estimated the upper limits on the redshifts. No convincing signal was foundand for each FRB we constrained the gamma-ray isotropic-equivalent luminosityand the released energy as a function of emission timescale. For the nearestFRB source, the periodic repeater FRB180916.J0158+65, we find$L_{\gamma,iso}<5.5\times 10^{47}$ erg/s over 1 s, whereas$L_{\gamma,iso}<10^{49}-10^{51}$ erg/s for the bulk of FRBs. The same valuesscale up by a factor of ~100 for a ms-long emission. Even on a timescalecomparable with that of the radio pulse itself no keV-MeV emission is observed.A systematic association with either long or short GRBs is ruled out with highconfidence, except for subluminous events, as is the case for core-collapse ofmassive stars (long) or binary neutron star mergers (short) viewed off axis.Only giant flares from extra-galactic magnetars at least ten times moreenergetic than Galactic siblings are ruled out for the nearest FRB.

A search for prompt gamma-ray counterparts to fast radio bursts in the Insight-HXMT data