We have compiled a catalog of 903 candidates for type 1 quasars at redshifts 3<z<5.5 selected among the X-ray sources of the serendipitous XMM-Newton survey presented in the 3XMM-DR4 catalog (the median X-ray flux is 5x10^{-15} erg/s/cm^2 the 0.5-2 keV energy band) and located at high Galactic latitudes >20 deg in Sloan Digital Sky Survey (SDSS) fields with a total area of about 300 deg^2. Photometric SDSS data as well infrared 2MASS and WISE data were used to select the objects. We selected the point sources from the photometric SDSS catalog with a magnitude error Delta z<0.2 and a color i-z<0.6 (to first eliminate the M-type stars). For the selected sources, we have calculated the dependences chi^2(z) for various spectral templates from the library that we compiled for these purposes using the EAZY software. Based on these data, we have rejected the objects whose spectral energy distributions are better described by the templates of stars at z=0 and obtained a sample of quasars with photometric redshift estimates 2.75<zphot<5.5. The selection completeness of known quasars at z>3 in the investigated fields is shown to be about 80%. The normalized median absolute deviation is 0.07, while the outlier fraction is eta= 9. The number of objects per unit area in our sample exceeds the number of quasars in the spectroscopic SDSS sample at the same redshifts approximately by a factor of 1.5. The subsequent spectroscopic testing of the redshifts of our selected candidates for quasars at 3<z<5.5 will allow the purity of this sample to be estimated more accurately.
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