Applying the recently constructed analytic representation for the pp scattering amplitudes, we present a study of p-air cross sections, with comparison to the data from Extensive Air Shower (EAS) measurements. The amplitudes describe with precision all available accelerator data at ISR, SPS and LHC energies, and its theoretical basis, together with the very smooth energy dependence of parameters controlled by unitarity and dispersion relations, permit reliable extrapolation to higher energies and to asymptotic ranges. The comparison with cosmic ray data is very satisfactory in the whole pp energy interval from 1 to 100 TeV. High energy asymptotic behaviour of cross sections is investigated in view of the geometric scaling property of the amplitudes. The amplitudes predict that the proton does not behave as a black disk even at asymptotically high enegies, and we discuss possible non-trivial consequences of this fact for pA collision cross sections at higher energies.
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