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Using the properties of the Euler's function, we found a way of calculating the number of preimages from large powers of two.  Also it was found a way to establish the exact number of preimages for a number of the form ${{2}^{{{2}^{n}}+a}},\,\,\,a<{{2}^{n}}$.  If the number ${{2}^{{{2}^{n}}+1}}$ is not prime. In the case if ${{2}^{{{2}^{n}}+1}}$ is  a simple one, a lower bound is found for the number of preimages of such a number.