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The energies and wave functions of the 1s, 2s, 2p(ml=0), and 2p(ml=±1) levels of hydrogen in a uniform magnetic field B (109 G≤B≤1012 G) are calculated in two cylindrical adiabatic approximations, each of which includes the influence of the Coulomb field on the radial motion of the electron. For the lowest levels of each symmetry [i.e., the 1s, 2p(ml=0) and 2p(ml=±1) levels] our calculations provide rigorously both upper and lower bounds on the true level energies and binding energies. We present these cylindrical adiabatic upper and lower bounds on the binding energies together with the spherical adiabatic upper and lower bounds of Starace and Webster as well as all superior variational lower bounds known to us in order to provide in one place stringent tests of past (and future) calculations. The very detailed 1984 eigenfunction expansion results of Rösner, Wunner, Herold, and Ruder are found to be consistent with the upper and lower bounds presented.