The knowledge of the dose and of the energy spectrum of the X-rays delivered to the patient during a radiological examination allows in principle the computation of the number of photons per unit surface useful for a good mammography. The film-screen assembly detectors used in the present standard practice require a number of photons per unit surface which, from a statistical point of view, would be unnecessarily high if single photon counting detectors with efficiency near to one were available. We discuss a possible solid state detector with these characteristics. Moreover, we propose the use of an X-ray monochromatic beam from a synchrotron radiation source in order to perform the examination at the energy where the signal to noise ratio has a maximum. Using the proposed detector in such a beam a substantial dose reduction can be foreseen.