  It is demonstrated here that if the prompt GRB emission is produced by the
simplest version of the external shock model, a specific relation should
prevail between the observed duration, isotropic equivalent energy, and photon
peak energy. In essence, this relation arises because both the burst duration
and the typical energy of the emitted synchrotron photons depend on the same
combination of the, usually poorly constrained, external density at the
deceleration radius, $n_{dec}$, and initial bulk Lorentz factor, $\Gamma_0$.
This has the fortunate consequence of making the relation independent of both
$\Gamma_0$ and $n_{dec}$. Unless the efficiency of electron acceleration is
very low, synchrotron gamma-rays from the external shock would fail to meet the
current observational constraints for the vast majority of GRBs, including
those with a smooth, single peak temporal profile. This argues either against
an external shock origin for the prompt emission in GRBs or for changes in our
understanding of the microphysical and radiation processes occurring within the
shocked region.
