Wien displacement law constant (frequency)

This is the constant behind the classic trap in blackbody radiation. The frequency at which a blackbody's spectral radiance peaks is ν_max = b′T, and it is not equal to c/λ_max. For the Sun, λ_max ≈ 502 nm implies 597 THz, while ν_max = b′T gives 339 THz — a factor of 1.76 apart, well into the near infrared.

There is no paradox. Spectral density is an amount of power per unit interval, and a fixed slice of wavelength is not a fixed slice of frequency: dν = (c/λ²) dλ, so changing variables reweights the curve and moves its maximum. The wavelength form solves x = 5(1 − e⁻ˣ) and the frequency form solves x = 3(1 − e⁻ˣ), giving 4.965 and 2.821 respectively. Always state which form you mean.