Hi Branko,
Let's try this again, with the tables in LaTeX...
Using NIST website CODATA values for 2010 and previous years, I have been looking at the accuracy of your formula. I wondered, using older experimental values, how well the formula could predict the currently best known value of mn/mp (i.e. from CODATA 2010).
First, here is the data I used:
[math]\begin{tabular}{l l l l}
Year &mass_e &mass_p &mass_n \\
\hline
1969 &9.1095580(540)e-31 kg &1.6726140(110)e-27 kg &1.6749200(110)e-27 kg\\
1973 &9.1095340(470)e-31 kg &1.6726485(86)e-27 kg &1.6749543(86)e-27 kg \\
1986 &9.10938970(540)e-31 kg &1.67262310(100)e-27 kg &1.67492860(100)e-27 kg\\
1998 &9.109381880(720)e-31 kg&1.672621580(130)e-27 kg &1.674927160(130)e-27 kg \\
2002 &9.10938260(160)e-31 kg &1.67262171(29)e-27 kg &1.67492728(29)e-27 kg\\
2006 &9.109382150(450)e-31 kg&1.672621637(83)e-27 kg &1.674927211(84)e-27 kg\\
2010 &9.109382910(400)e-31 kg&1.672621777(74)e-27 kg &1.674927351(74)e-27 kg\\
\end{tabular}[/math]
[math]\begin{tabular}{l l l l}
Year &1/alpha &mass_p/mass_e &mass_n/mass_p\\
\hline
1969 &137.03602(21) &1836.1090(110) &1.001379(13)\\
1973 &137.036040(110) &1836.15152(70) &1.001379(10)\\
1986 &137.0359895(61) &1836.152701(37) &1.0013784040(90)\\
1998 &137.03599976(50) &1836.1526675(39) &1.00137841887(58)\\
2002 &137.03599911(46) &1836.152667261(85) &1.00137841870(58)\\
2006 &137.035999679(94) &1836.152667247(80) &1.00137841918(46)\\
2010 &137.035999074(45) &1836.152667245(75) &1.00137841917(45)\\
\end{tabular}[/math]
In 1969 and 1973, the ratio mass_n/mass_p is not reported separately, so it is calculated from the two mass values given.
In the next table, the "CODATA gamma" is the value given in the particular year. The "Zivlak gamma" is the value obtained by using the experimental values known at the time in the Zivlak equation.
[math]\begin{tabular}{l l l}
Year &CODATA \gamma &Zivlak \gamma \\
\hline
1969 &1.001379(13) &1.0013784178(53)\\
1973 &1.001379(10) &1.0013784162(23)\\
1986 &1.0013784040(90) &1.00137841927(13)\\
1998 &1.00137841887(58) &1.001378419181(10)\\
2002 &1.00137841870(58) &1.0013784191948(93)\\
2006 &1.00137841918(46) &1.0013784191907(19)\\
2010 &1.00137841917(45) &1.00137841920390(92)\\
\end{tabular}[/math]
We now look at the success of the Zivlak equation in predicting the current value (CODATA 2010) for gamma.
The "CODATA Error" is the difference between the value given at the time (i.e. in 1969 and so on) and the current 2010 value, as a proportion of the 2010 value.
The "Zivlak error" similarily, is the difference between the value that could have been calculated at the time with the current 2010 value, as a proportion of the 2010 value.
The C Error/Z Error column shows the ratio of the CODATA error to the Zivlak error.
[math]\begin{tabular}{l l l l}
Year &CODATA error &Zivlak Error &C Error/Z Error\\
\hline
1969 &1.3144463E-5 &5.790777E-9 &2269.8962\\
1973 &1.0276465E-5 &2.7340497E-9 &3758.6973\\
1986 &1.5149118E-8 &5.758918E-10 &26.305494\\
1998 &1.0285822E-9 &4.5983836E-10 &2.2368343\\
2002 &1.0301803E-9 &4.586465E-10 &2.2461312\\
2006 &9.100147E-10 &4.5128046E-10 &2.016517\\
2010 &8.9876123E-10 &4.5029516E-10 &1.995938\\
\end{tabular}[/math]
There are several things of note.
(1) First, note that, for all years, the Zivlak gamma is significantly closer to the current known value than the value that was obtained at the time via the sophisticated methods of CODATA. For example, in 1973, it was 3758 times more accurate (this is partially due to the fact that the mn/mp ratio was not disclosed by CODATA).
(2) By 1986, the Zivlak equation had produced a value that is more accurate than the value we have from CODATA even today.
(3) Perhaps most surprisingly, even when we compare the 2010 CODATA value against itself as the gold standard, the Zivlak value is superior. This is because it predicts a (slightly different but very precise) value with very little uncertainty. The larger uncertainty in the 2010 CODATA value means its average error is higher.
These results suggest that the Zivlak formula for the ratio of neutron to proton mass has real predictive power. Please accept my congratulations for your work on this!
Hugh