Risaliti and Lusso have compiled X-ray and UV flux measurements of 1598 quasars (QSOs) in the redshift range $0.036 leq z leq 5.1003$, part of which, $z sim 2.4 - 5.1$, is largely cosmologically unprobed. In this paper we use these QSO measurements, alone and in conjunction with baryon acoustic oscillation (BAO) and Hubble parameter [$H(z)$] measurements, to constrain cosmological parameters in six different cosmological models, each with two different Hubble constant priors. In most of these models, given the larger uncertainties, the QSO cosmological parameter constraints are mostly consistent with those from the $H(z)$ + BAO data. A somewhat significant exception is the non-relativistic matter density parameter $Omega_{m0}$ where the QSO data favors $Omega_{m0} sim 0.5 - 0.6$ in most models. Consequently in joint analyses of QSO data with $H(z)$ + BAO data the one-dimensional $Omega_{m0}$ distributions shift slightly toward larger values. A joint analysis of the QSO + $H(z)$ + BAO data is consistent with the current standard model, spatially-flat $Lambda$CDM, but mildly favors closed spatial hypersurfaces and dynamical dark energy. Since the higher $Omega_{m0}$ values favored by the QSO data appear to be associated with the $z sim 2 - 5$ part of these data, and conflict somewhat with strong indications for $Omega_{m0} sim 0.3$ from most $z < 2.5$ data as well as from the cosmic microwave background anisotropy data at $z sim 1100$, in most models, the larger QSO data $Omega_{m0}$ is possibly more indicative of an issue with the $z sim 2 - 5$ QSO data than of an inadequacy of the standard flat $Lambda$CDM model.
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