No robust detection of prompt electromagnetic counterparts to fast radio bursts (FRBs) has yet been obtained, in spite of several multi-wavelength searches carried out so far. Specifically, X/gamma-ray counterparts are predicted by some models. We planned on searching for prompt gamma-ray counterparts in the Insight-Hard X-ray Modulation Telescope (Insight-HXMT) data, taking advantage of the unique combination of large effective area in the keV-MeV energy range and of sub-ms time resolution. We selected 39 FRBs that were 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 time profiles over a wide range of time resolutions, from several seconds down to sub-ms scales. Using the dispersion measures in excess of the Galactic terms, we estimated the upper limits on the redshifts. No convincing signal was found and for each FRB we constrained the gamma-ray isotropic-equivalent luminosity and the released energy as a function of emission timescale. For the nearest FRB source, the periodic repeater FRB180916.J0158+65, we find $L_{gamma,iso}<5.5times 10^{47}$ erg/s over 1 s, whereas $L_{gamma,iso}<10^{49}-10^{51}$ erg/s for the bulk of FRBs. The same values scale up by a factor of ~100 for a ms-long emission. Even on a timescale comparable 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 high confidence, except for subluminous events, as is the case for core-collapse of massive stars (long) or binary neutron star mergers (short) viewed off axis. Only giant flares from extra-galactic magnetars at least ten times more energetic than Galactic siblings are ruled out for the nearest FRB.
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