The total nucleon-nucleon cross section at large N_c

It is shown that at sufficiently large $N_c$ for incident momenta which are much larger than the QCD, the total nucleon-nucleon cross section is independent of incident momentum and given by $\sigma^{\rm total}=2 \pi \log^2(N_c) / (m^2_{\pi})$. This result is valid in the extreme large $N_c$ regime of $\log(N_c) \gg 1$ and has corrections of relative order $\log (\log(N_c))/\log(N_c)$. A possible connection of this result to the Froissart-Martin bound is discussed.