= Solution
Density parameters at the same epoch share a common critical-density denominator. Thus compare the densities directly: $\rho_m=\rho_{m0}(1+z)^3$ and $\rho_\gamma=\rho_{\gamma0}(1+z)^4$. Equality gives
$$
\boxed{1+z_{m\gamma}=\frac{\Omega_{m0}}{\Omega_{\gamma0}}=6245.89,\qquad z_{m\gamma}\simeq6244.89\simeq6.24\times10^3}.
$$
This is the <photon-only matter-radiation equality> appropriate to the specified radiation density, not the equality including all relativistic species.
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