= Solution
For a $5$ GeV electron, $\gamma_e=E/(m_ec^2)\simeq9785$. With $\rho=10$ m, the stated <synchrotron critical frequency> formula gives
$$
\boxed{\epsilon_{c,e}\simeq2.77\times10^4\,\mathrm{eV}=27.7\,\mathrm{keV}.}
$$
At the same total beam energy and bend radius, a proton has $\gamma_p=5/0.9383\simeq5.329$, and the same supplied formula gives
$$
\boxed{\epsilon_{c,p}\simeq4.48\times10^{-6}\,\mathrm{eV}.}
$$
The ratio is $(\gamma_p/\gamma_e)^3\simeq(m_e/m_p)^3$, explaining the strong suppression for protons. The critical-energy formula is an ultrarelativistic approximation, excellent for the electron and only approximate for this moderately relativistic proton. If “5 GeV proton beam” instead denotes kinetic energy, then $\gamma_p=1+5/0.9383$ and the estimate is about $7.50\times10^{-6}$ eV; the two beam-energy conventions should not be mixed.
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