Rydberg Function
$$R_H\equiv{m_em_p\over{m_e+m_p}}{e^4\over8c\epsilon_0^2h^3}\approx{m_e}{e^4\over8c\epsilon_0^2h^3}$$
$$R_H\equiv{m_em_p\over{m_e+m_p}}{e^4\over8c\epsilon_0^2h^3}={m_e\over{1+{m_e\over m_p}}}{e^4\over8c\epsilon_0^2h^3}$$
$$R_H\equiv{m_em_p\over{m_e+m_p}}{e^4\over8c\epsilon_0^2h^3}={m_e\over{1+{m_e\over m_p}}}{e^4\over8c\epsilon_0^2h^3}$$
$$1\equiv{m_e\over{1+{m_e\over m_p}}}{e^4\over8c\epsilon_0^2h^3R_H}$$
$$1+{m_e\over m_p}={m_e}{e^4\over8c\epsilon_0^2h^3R_H}$$
$$F\left(x,\cdots,x_n\right)\equiv1\equiv{m_e}{e^4\over8c\epsilon_0^2h^3R_H}-{m_e\over m_p}\approx{m_e}{e^4\over8c\epsilon_0^2h^3R_H}$$
$$F\left(x,\cdots,x_n\right)\equiv1\equiv{m_e}{e^4\over8c\epsilon_0^2h^3R_H}-{{\pi r_pcm_e}\over2h}\approx{m_e}{e^4\over8c\epsilon_0^2h^3R_H}$$
The roots of this multi-dimensional polynomial are the complete solution to the proton radius problem and all of standard physics.
The roots of this multi-dimensional polynomial are the complete solution to the proton radius problem and all of standard physics.
This approximation
${np\over{n+p}}= Constant \approx n$ if $p\gg n $ approximation is used in many fields where the product of two parameters is a constant. It is used often in derivations. Misuse is a big problem when comparing theory to measurement. (!!!) (this may be root of confusion)
Likely one of these equations converges numerically, and the other is challenging.
I suspect this approximation is related to the major blunder I heard/sensed rippling through the corporate scientific researcher community in the late 1980s, early 1990s.
