  It is demonstrated that a uniform magnetic field can exactly pair the
two-dimensional (2D) charged particles only for some quantized magnetic
intensity values. For the particle-pair consisting of two like charges the
Landau level of the center-of-mass motion is multiple degenerate that implies
the shell structure. However, any particle-pair has only two non-degenerate
relative levels, which are associated with the diamagnetic and paramagnetic
states respectively. There exist a upper critical magnetic strength and a lower
critical magnetic length across which two like charges cannot be paired. The
theoretical results agree with the experimental data on the sites and widths of
the integral and fractional quantum Hall plateaus in a CaAs-Al$_x$Ga$_{1-x}$As
heterojunction, that gives a new explanation for the quantum Hall effect.
