Peak CMB Wavelength from the Hubble Parameter and the Planck Scale and the Planck Length from Peak CMB Wavelength

The two commonly quoted peak wavelengths of the Planck blackbody spectrum---the maximum of the wavelength spectral radiance $B_\lambda$ and the wavelength corresponding to the maximum of the frequency spectral radiance $B_\nu$---are combined with a proposed relation between the cosmic microwave background (CMB) temperature, the Hubble parameter, and the Planck length. This gives an exact expression for the peak CMB wavelength in terms of the Hubble radius and the Planck scale, including the Lambert-$W$ solutions of the two transcendental peak equations. Their exact ratio is obtained and both results are evaluated first using the Particle Data Group value of the Hubble constant and then using the Hubble constant reported by Haug and Tatum from the PantheonPlusSH0ES Type Ia supernova database. The latter comparison is explicitly identified as a consistency check rather than an independent prediction, because the Fixsen CMB temperature was used as an input in deriving that value of the Hubble constant. The relation is also inverted to express the Planck length in terms of the observed peak wavelength, illustrating a conditional route to the Planck length without explicit input of Newton's gravitational constant.